9 25

9_24

9_26

Visit 9 25's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 25's page at Knotilus! Visit 9 25's page at the original Knot Atlas! 9 25 Quick Notes

9 25 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X5,12,6,13 X9,17,10,16 X13,18,14,1 X17,14,18,15 X15,11,16,10 X11,6,12,7 X7283
Gauss code	-1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -8, 3, -5, 6, -7, 4, -6, 5
Dowker-Thistlethwaite code	4 8 12 2 16 6 18 10 14
Conway Notation	[22,21,2]

Minimum Braid Representative: Length is 10, width is 5. Braid index is 5.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-10][-1] Hyperbolic Volume 11.3903 A-Polynomial See Data:9 25/A-polynomial

[edit Notes for 9 25's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant -2

[edit Notes for 9 25's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+12t-17+12t^{-1}-3t^{-2}$
Conway polynomial	$1-3z^{4}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 47, -2 }
Jones polynomial	$q-2+5q^{-1}-7q^{-2}+8q^{-3}-8q^{-4}+7q^{-5}-5q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{8}+3z^{2}a^{6}+3a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-3a^{4}-z^{4}a^{2}+a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-7z^{4}a^{8}+4z^{2}a^{8}-a^{8}+3z^{7}a^{7}-4z^{5}a^{7}-2z^{3}a^{7}+za^{7}+z^{8}a^{6}+6z^{6}a^{6}-18z^{4}a^{6}+13z^{2}a^{6}-3a^{6}+6z^{7}a^{5}-10z^{5}a^{5}+5z^{3}a^{5}-za^{5}+z^{8}a^{4}+6z^{6}a^{4}-15z^{4}a^{4}+13z^{2}a^{4}-3a^{4}+3z^{7}a^{3}-3z^{5}a^{3}+3z^{3}a^{3}-za^{3}+3z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}-a^{2}+2z^{5}a-2z^{3}a+z^{4}-2z^{2}+1$
The A2 invariant	$-q^{26}-q^{24}+2q^{22}+q^{18}+2q^{16}-2q^{14}-2q^{10}+q^{6}-q^{4}+3q^{2}+q^{-4}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+7q^{120}-4q^{118}-6q^{116}+19q^{114}-29q^{112}+34q^{110}-28q^{108}+4q^{106}+23q^{104}-50q^{102}+63q^{100}-55q^{98}+26q^{96}+12q^{94}-46q^{92}+60q^{90}-48q^{88}+22q^{86}+15q^{84}-38q^{82}+41q^{80}-18q^{78}-14q^{76}+48q^{74}-60q^{72}+50q^{70}-13q^{68}-30q^{66}+68q^{64}-87q^{62}+79q^{60}-45q^{58}-6q^{56}+48q^{54}-78q^{52}+76q^{50}-50q^{48}+8q^{46}+26q^{44}-46q^{42}+38q^{40}-14q^{38}-20q^{36}+43q^{34}-43q^{32}+21q^{30}+13q^{28}-44q^{26}+63q^{24}-54q^{22}+31q^{20}-q^{18}-27q^{16}+43q^{14}-43q^{12}+34q^{10}-14q^{8}+11q^{4}-16q^{2}+15-11q^{-2}+8q^{-4}-2q^{-6}-q^{-8}+3q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 9_25.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+2q^{15}-2q^{13}+2q^{11}-q^{9}+q^{5}-2q^{3}+3q-q^{-1}+q^{-3}$
2	$q^{48}-2q^{46}-2q^{44}+7q^{42}-2q^{40}-9q^{38}+11q^{36}+3q^{34}-15q^{32}+7q^{30}+8q^{28}-12q^{26}+q^{24}+8q^{22}-3q^{20}-6q^{18}+3q^{16}+9q^{14}-10q^{12}-3q^{10}+16q^{8}-8q^{6}-8q^{4}+12q^{2}-2-5q^{-2}+4q^{-4}-q^{-8}+q^{-10}$
3	$-q^{93}+2q^{91}+2q^{89}-3q^{87}-7q^{85}+2q^{83}+16q^{81}+q^{79}-23q^{77}-12q^{75}+29q^{73}+29q^{71}-29q^{69}-46q^{67}+19q^{65}+59q^{63}-2q^{61}-68q^{59}-15q^{57}+66q^{55}+30q^{53}-55q^{51}-40q^{49}+44q^{47}+44q^{45}-29q^{43}-44q^{41}+11q^{39}+40q^{37}+7q^{35}-36q^{33}-28q^{31}+29q^{29}+46q^{27}-17q^{25}-61q^{23}+3q^{21}+70q^{19}+14q^{17}-69q^{15}-25q^{13}+55q^{11}+36q^{9}-39q^{7}-35q^{5}+24q^{3}+27q-8q^{-1}-18q^{-3}+3q^{-5}+9q^{-7}-q^{-9}-4q^{-11}+q^{-13}+q^{-15}-q^{-19}+q^{-21}$
4	$q^{152}-2q^{150}-2q^{148}+3q^{146}+3q^{144}+7q^{142}-9q^{140}-16q^{138}-q^{136}+11q^{134}+41q^{132}-q^{130}-47q^{128}-44q^{126}-10q^{124}+101q^{122}+72q^{120}-33q^{118}-123q^{116}-129q^{114}+98q^{112}+190q^{110}+105q^{108}-121q^{106}-295q^{104}-53q^{102}+211q^{100}+299q^{98}+33q^{96}-349q^{94}-252q^{92}+79q^{90}+374q^{88}+218q^{86}-243q^{84}-338q^{82}-85q^{80}+301q^{78}+291q^{76}-93q^{74}-289q^{72}-165q^{70}+170q^{68}+258q^{66}+32q^{64}-191q^{62}-191q^{60}+36q^{58}+196q^{56}+152q^{54}-78q^{52}-215q^{50}-129q^{48}+113q^{46}+288q^{44}+76q^{42}-208q^{40}-305q^{38}-32q^{36}+357q^{34}+254q^{32}-95q^{30}-388q^{28}-210q^{26}+269q^{24}+329q^{22}+82q^{20}-286q^{18}-287q^{16}+78q^{14}+232q^{12}+171q^{10}-100q^{8}-202q^{6}-36q^{4}+74q^{2}+119+7q^{-2}-73q^{-4}-33q^{-6}-2q^{-8}+40q^{-10}+13q^{-12}-14q^{-14}-4q^{-16}-7q^{-18}+7q^{-20}+2q^{-22}-3q^{-24}+2q^{-26}-2q^{-28}+q^{-30}-q^{-34}+q^{-36}$
5	$-q^{225}+2q^{223}+2q^{221}-3q^{219}-3q^{217}-3q^{215}+9q^{211}+16q^{209}+q^{207}-20q^{205}-29q^{203}-19q^{201}+19q^{199}+61q^{197}+65q^{195}-8q^{193}-97q^{191}-128q^{189}-59q^{187}+101q^{185}+232q^{183}+192q^{181}-50q^{179}-314q^{177}-380q^{175}-132q^{173}+317q^{171}+608q^{169}+433q^{167}-179q^{165}-764q^{163}-819q^{161}-168q^{159}+769q^{157}+1205q^{155}+673q^{153}-549q^{151}-1463q^{149}-1236q^{147}+92q^{145}+1502q^{143}+1753q^{141}+499q^{139}-1299q^{137}-2079q^{135}-1103q^{133}+880q^{131}+2172q^{129}+1600q^{127}-375q^{125}-2039q^{123}-1894q^{121}-105q^{119}+1723q^{117}+1983q^{115}+488q^{113}-1347q^{111}-1887q^{109}-723q^{107}+973q^{105}+1668q^{103}+846q^{101}-645q^{99}-1424q^{97}-882q^{95}+374q^{93}+1185q^{91}+903q^{89}-124q^{87}-979q^{85}-964q^{83}-132q^{81}+798q^{79}+1067q^{77}+447q^{75}-597q^{73}-1215q^{71}-837q^{69}+336q^{67}+1349q^{65}+1278q^{63}+33q^{61}-1400q^{59}-1722q^{57}-511q^{55}+1301q^{53}+2085q^{51}+1038q^{49}-1011q^{47}-2255q^{45}-1542q^{43}+537q^{41}+2191q^{39}+1913q^{37}-10q^{35}-1840q^{33}-2044q^{31}-521q^{29}+1324q^{27}+1924q^{25}+878q^{23}-749q^{21}-1559q^{19}-1024q^{17}+235q^{15}+1098q^{13}+964q^{11}+102q^{9}-649q^{7}-735q^{5}-264q^{3}+292q+489q^{-1}+271q^{-3}-86q^{-5}-266q^{-7}-199q^{-9}-12q^{-11}+122q^{-13}+116q^{-15}+34q^{-17}-43q^{-19}-59q^{-21}-23q^{-23}+13q^{-25}+20q^{-27}+12q^{-29}+3q^{-31}-10q^{-33}-6q^{-35}+3q^{-37}+3q^{-43}-q^{-45}-2q^{-47}+q^{-49}-q^{-53}+q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{26}-q^{24}+2q^{22}+q^{18}+2q^{16}-2q^{14}-2q^{10}+q^{6}-q^{4}+3q^{2}+q^{-4}$
1,1	$q^{68}-4q^{66}+12q^{64}-28q^{62}+52q^{60}-86q^{58}+130q^{56}-176q^{54}+212q^{52}-234q^{50}+232q^{48}-194q^{46}+126q^{44}-30q^{42}-80q^{40}+196q^{38}-303q^{36}+380q^{34}-432q^{32}+438q^{30}-410q^{28}+342q^{26}-244q^{24}+138q^{22}-19q^{20}-76q^{18}+154q^{16}-198q^{14}+214q^{12}-206q^{10}+174q^{8}-142q^{6}+107q^{4}-74q^{2}+50-30q^{-2}+21q^{-4}-10q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$
2,0	$q^{66}+q^{64}-q^{62}-4q^{60}-2q^{58}+4q^{56}+2q^{54}-3q^{52}+8q^{48}+5q^{46}-8q^{44}-5q^{42}+4q^{40}-q^{38}-8q^{36}-3q^{34}+6q^{32}+q^{30}-q^{28}+4q^{26}+q^{24}-2q^{22}+5q^{20}+4q^{18}-7q^{16}-2q^{14}+7q^{12}+q^{10}-9q^{8}-2q^{6}+9q^{4}-5+q^{-2}+4q^{-4}+q^{-6}-q^{-8}+q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}-2q^{52}+q^{50}+3q^{48}-7q^{46}+3q^{44}+4q^{42}-11q^{40}+6q^{38}+7q^{36}-10q^{34}+4q^{32}+7q^{30}-5q^{28}-q^{26}+3q^{24}+2q^{22}-4q^{20}-4q^{18}+8q^{16}-5q^{14}-8q^{12}+12q^{10}-3q^{8}-7q^{6}+10q^{4}-3+4q^{-2}+q^{-4}-q^{-6}+q^{-8}$
1,0,0	$-q^{35}-q^{33}-q^{31}+2q^{29}+3q^{25}+q^{23}+2q^{21}-2q^{19}-q^{17}-2q^{15}-2q^{13}+2q^{7}-q^{5}+3q^{3}+q^{-1}+q^{-5}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+2q^{52}-5q^{50}+7q^{48}-9q^{46}+11q^{44}-12q^{42}+11q^{40}-8q^{38}+5q^{36}+2q^{34}-6q^{32}+13q^{30}-17q^{28}+21q^{26}-23q^{24}+20q^{22}-18q^{20}+12q^{18}-8q^{16}+q^{14}+4q^{12}-8q^{10}+11q^{8}-11q^{6}+12q^{4}-8q^{2}+7-4q^{-2}+3q^{-4}-q^{-6}+q^{-8}$
1,0	$q^{88}-2q^{84}-2q^{82}+3q^{80}+5q^{78}-2q^{76}-8q^{74}-3q^{72}+9q^{70}+8q^{68}-7q^{66}-12q^{64}+q^{62}+13q^{60}+6q^{58}-10q^{56}-8q^{54}+5q^{52}+9q^{50}-2q^{48}-8q^{46}+8q^{42}+2q^{40}-8q^{38}-3q^{36}+7q^{34}+6q^{32}-6q^{30}-8q^{28}+4q^{26}+10q^{24}-2q^{22}-12q^{20}-3q^{18}+11q^{16}+8q^{14}-7q^{12}-11q^{10}+q^{8}+11q^{6}+4q^{4}-4q^{2}-5+q^{-2}+4q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-14}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-2q^{126}+5q^{124}-8q^{122}+7q^{120}-4q^{118}-6q^{116}+19q^{114}-29q^{112}+34q^{110}-28q^{108}+4q^{106}+23q^{104}-50q^{102}+63q^{100}-55q^{98}+26q^{96}+12q^{94}-46q^{92}+60q^{90}-48q^{88}+22q^{86}+15q^{84}-38q^{82}+41q^{80}-18q^{78}-14q^{76}+48q^{74}-60q^{72}+50q^{70}-13q^{68}-30q^{66}+68q^{64}-87q^{62}+79q^{60}-45q^{58}-6q^{56}+48q^{54}-78q^{52}+76q^{50}-50q^{48}+8q^{46}+26q^{44}-46q^{42}+38q^{40}-14q^{38}-20q^{36}+43q^{34}-43q^{32}+21q^{30}+13q^{28}-44q^{26}+63q^{24}-54q^{22}+31q^{20}-q^{18}-27q^{16}+43q^{14}-43q^{12}+34q^{10}-14q^{8}+11q^{4}-16q^{2}+15-11q^{-2}+8q^{-4}-2q^{-6}-q^{-8}+3q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 25"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-3t^{2}+12t-17+12t^{-1}-3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$1-3z^{4}$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 47, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q-2+5q^{-1}-7q^{-2}+8q^{-3}-8q^{-4}+7q^{-5}-5q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-a^{8}+3z^{2}a^{6}+3a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-3a^{4}-z^{4}a^{2}+a^{2}+z^{2}+1$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-7z^{4}a^{8}+4z^{2}a^{8}-a^{8}+3z^{7}a^{7}-4z^{5}a^{7}-2z^{3}a^{7}+za^{7}+z^{8}a^{6}+6z^{6}a^{6}-18z^{4}a^{6}+13z^{2}a^{6}-3a^{6}+6z^{7}a^{5}-10z^{5}a^{5}+5z^{3}a^{5}-za^{5}+z^{8}a^{4}+6z^{6}a^{4}-15z^{4}a^{4}+13z^{2}a^{4}-3a^{4}+3z^{7}a^{3}-3z^{5}a^{3}+3z^{3}a^{3}-za^{3}+3z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}-a^{2}+2z^{5}a-2z^{3}a+z^{4}-2z^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n134, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n25, ...}

Vassiliev invariants

V2 and V3:

(0, -1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $0$ $-8$ $0$ $64$ $24$ $0$ $-{\frac {656}{3}}$ $-{\frac {128}{3}}$ $-72$ $0$ $32$ $0$ $0$ $528$ $-{\frac {248}{3}}$ $344$ $48$ $32$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-2 is the signature of 9 25. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   1 1 1                 1   -1 -1               4 1   3 -3             4 2     -2 -5           4 3       1 -7         4 4         0 -9       3 4           -1 -11     2 4             2 -13   1 3               -2 -15   2                 2 -17 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-2 q^3+q^2+5 q-11+4 q^{-1} +19 q^{-2} -31 q^{-3} +4 q^{-4} +43 q^{-5} -50 q^{-6} -3 q^{-7} +62 q^{-8} -56 q^{-9} -12 q^{-10} +65 q^{-11} -45 q^{-12} -19 q^{-13} +52 q^{-14} -25 q^{-15} -20 q^{-16} +30 q^{-17} -7 q^{-18} -12 q^{-19} +10 q^{-20} -3 q^{-22} + q^{-23} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^9-2 q^8+q^7+q^6+q^5-7 q^4+4 q^3+11 q^2-5 q-28+14 q^{-1} +46 q^{-2} -8 q^{-3} -87 q^{-4} +10 q^{-5} +121 q^{-6} +11 q^{-7} -167 q^{-8} -34 q^{-9} +204 q^{-10} +67 q^{-11} -234 q^{-12} -98 q^{-13} +248 q^{-14} +130 q^{-15} -251 q^{-16} -155 q^{-17} +240 q^{-18} +173 q^{-19} -218 q^{-20} -184 q^{-21} +185 q^{-22} +188 q^{-23} -145 q^{-24} -184 q^{-25} +101 q^{-26} +173 q^{-27} -60 q^{-28} -148 q^{-29} +20 q^{-30} +120 q^{-31} +6 q^{-32} -87 q^{-33} -20 q^{-34} +55 q^{-35} +23 q^{-36} -29 q^{-37} -20 q^{-38} +14 q^{-39} +12 q^{-40} -5 q^{-41} -5 q^{-42} +3 q^{-44} - q^{-45} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{16}-2 q^{15}+q^{14}+q^{13}-3 q^{12}+5 q^{11}-7 q^{10}+6 q^9+6 q^8-17 q^7+8 q^6-17 q^5+33 q^4+33 q^3-59 q^2-23 q-57+113 q^{-1} +145 q^{-2} -104 q^{-3} -133 q^{-4} -223 q^{-5} +215 q^{-6} +416 q^{-7} -43 q^{-8} -287 q^{-9} -588 q^{-10} +216 q^{-11} +784 q^{-12} +204 q^{-13} -347 q^{-14} -1067 q^{-15} +38 q^{-16} +1077 q^{-17} +553 q^{-18} -244 q^{-19} -1456 q^{-20} -235 q^{-21} +1174 q^{-22} +837 q^{-23} -32 q^{-24} -1631 q^{-25} -477 q^{-26} +1088 q^{-27} +974 q^{-28} +198 q^{-29} -1587 q^{-30} -637 q^{-31} +861 q^{-32} +974 q^{-33} +421 q^{-34} -1361 q^{-35} -725 q^{-36} +526 q^{-37} +850 q^{-38} +617 q^{-39} -977 q^{-40} -715 q^{-41} +140 q^{-42} +597 q^{-43} +712 q^{-44} -516 q^{-45} -559 q^{-46} -155 q^{-47} +266 q^{-48} +615 q^{-49} -134 q^{-50} -293 q^{-51} -243 q^{-52} +2 q^{-53} +373 q^{-54} +40 q^{-55} -67 q^{-56} -158 q^{-57} -90 q^{-58} +146 q^{-59} +46 q^{-60} +23 q^{-61} -53 q^{-62} -61 q^{-63} +35 q^{-64} +12 q^{-65} +20 q^{-66} -7 q^{-67} -19 q^{-68} +5 q^{-69} +5 q^{-71} -3 q^{-73} + q^{-74} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{25}-2 q^{24}+q^{23}+q^{22}-3 q^{21}+q^{20}+5 q^{19}-5 q^{18}+q^{17}+4 q^{16}-12 q^{15}-3 q^{14}+18 q^{13}+4 q^{12}+9 q^{11}-3 q^{10}-48 q^9-39 q^8+34 q^7+81 q^6+91 q^5+3 q^4-182 q^3-226 q^2-33 q+261+448 q^{-1} +221 q^{-2} -379 q^{-3} -782 q^{-4} -504 q^{-5} +347 q^{-6} +1199 q^{-7} +1083 q^{-8} -245 q^{-9} -1645 q^{-10} -1763 q^{-11} -188 q^{-12} +2009 q^{-13} +2710 q^{-14} +801 q^{-15} -2245 q^{-16} -3608 q^{-17} -1711 q^{-18} +2213 q^{-19} +4540 q^{-20} +2724 q^{-21} -1967 q^{-22} -5262 q^{-23} -3790 q^{-24} +1500 q^{-25} +5784 q^{-26} +4773 q^{-27} -920 q^{-28} -6046 q^{-29} -5602 q^{-30} +289 q^{-31} +6106 q^{-32} +6206 q^{-33} +325 q^{-34} -5975 q^{-35} -6615 q^{-36} -884 q^{-37} +5728 q^{-38} +6824 q^{-39} +1369 q^{-40} -5355 q^{-41} -6884 q^{-42} -1814 q^{-43} +4896 q^{-44} +6809 q^{-45} +2224 q^{-46} -4328 q^{-47} -6602 q^{-48} -2625 q^{-49} +3640 q^{-50} +6267 q^{-51} +3003 q^{-52} -2837 q^{-53} -5780 q^{-54} -3320 q^{-55} +1944 q^{-56} +5103 q^{-57} +3543 q^{-58} -1002 q^{-59} -4285 q^{-60} -3580 q^{-61} +116 q^{-62} +3314 q^{-63} +3398 q^{-64} +662 q^{-65} -2310 q^{-66} -3008 q^{-67} -1176 q^{-68} +1331 q^{-69} +2422 q^{-70} +1442 q^{-71} -512 q^{-72} -1754 q^{-73} -1427 q^{-74} -79 q^{-75} +1094 q^{-76} +1215 q^{-77} +402 q^{-78} -532 q^{-79} -895 q^{-80} -515 q^{-81} +157 q^{-82} +564 q^{-83} +457 q^{-84} +53 q^{-85} -283 q^{-86} -340 q^{-87} -134 q^{-88} +115 q^{-89} +209 q^{-90} +119 q^{-91} -19 q^{-92} -98 q^{-93} -94 q^{-94} -16 q^{-95} +49 q^{-96} +50 q^{-97} +12 q^{-98} -9 q^{-99} -21 q^{-100} -20 q^{-101} +7 q^{-102} +12 q^{-103} +2 q^{-104} -5 q^{-107} +3 q^{-109} - q^{-110} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}-2 q^{35}+q^{34}+q^{33}-3 q^{32}+q^{31}+q^{30}+7 q^{29}-10 q^{28}-q^{27}+9 q^{26}-13 q^{25}+q^{24}+9 q^{23}+28 q^{22}-24 q^{21}-19 q^{20}+13 q^{19}-46 q^{18}-4 q^{17}+51 q^{16}+119 q^{15}-15 q^{14}-69 q^{13}-52 q^{12}-220 q^{11}-82 q^{10}+166 q^9+468 q^8+248 q^7-36 q^6-281 q^5-904 q^4-671 q^3+129 q^2+1296 q+1406+843 q^{-1} -286 q^{-2} -2463 q^{-3} -2814 q^{-4} -1280 q^{-5} +2022 q^{-6} +3948 q^{-7} +4080 q^{-8} +1692 q^{-9} -3997 q^{-10} -7111 q^{-11} -6048 q^{-12} +341 q^{-13} +6591 q^{-14} +10268 q^{-15} +7942 q^{-16} -2671 q^{-17} -11812 q^{-18} -14534 q^{-19} -6166 q^{-20} +6068 q^{-21} +17110 q^{-22} +18345 q^{-23} +3858 q^{-24} -13313 q^{-25} -23788 q^{-26} -16818 q^{-27} +302 q^{-28} +20740 q^{-29} +29289 q^{-30} +14325 q^{-31} -9763 q^{-32} -29726 q^{-33} -27651 q^{-34} -8996 q^{-35} +19526 q^{-36} +36660 q^{-37} +24647 q^{-38} -2982 q^{-39} -30850 q^{-40} -34893 q^{-41} -17980 q^{-42} +15180 q^{-43} +39209 q^{-44} +31596 q^{-45} +3769 q^{-46} -28659 q^{-47} -37739 q^{-48} -24145 q^{-49} +10297 q^{-50} +38309 q^{-51} +34815 q^{-52} +8783 q^{-53} -25138 q^{-54} -37536 q^{-55} -27577 q^{-56} +5901 q^{-57} +35503 q^{-58} +35603 q^{-59} +12527 q^{-60} -20802 q^{-61} -35488 q^{-62} -29521 q^{-63} +1281 q^{-64} +30969 q^{-65} +34848 q^{-66} +16129 q^{-67} -14860 q^{-68} -31473 q^{-69} -30505 q^{-70} -4448 q^{-71} +23891 q^{-72} +32060 q^{-73} +19627 q^{-74} -6779 q^{-75} -24549 q^{-76} -29514 q^{-77} -10684 q^{-78} +14058 q^{-79} +25993 q^{-80} +21279 q^{-81} +2141 q^{-82} -14620 q^{-83} -24835 q^{-84} -14902 q^{-85} +3352 q^{-86} +16539 q^{-87} +18845 q^{-88} +8669 q^{-89} -3934 q^{-90} -16341 q^{-91} -14471 q^{-92} -4504 q^{-93} +6245 q^{-94} +12251 q^{-95} +10001 q^{-96} +3585 q^{-97} -6897 q^{-98} -9567 q^{-99} -6822 q^{-100} -880 q^{-101} +4656 q^{-102} +6712 q^{-103} +5644 q^{-104} -498 q^{-105} -3620 q^{-106} -4667 q^{-107} -2986 q^{-108} -165 q^{-109} +2389 q^{-110} +3804 q^{-111} +1480 q^{-112} -58 q^{-113} -1619 q^{-114} -1905 q^{-115} -1366 q^{-116} -17 q^{-117} +1418 q^{-118} +950 q^{-119} +732 q^{-120} -54 q^{-121} -510 q^{-122} -820 q^{-123} -452 q^{-124} +252 q^{-125} +207 q^{-126} +391 q^{-127} +194 q^{-128} +39 q^{-129} -248 q^{-130} -225 q^{-131} +3 q^{-132} -31 q^{-133} +92 q^{-134} +80 q^{-135} +76 q^{-136} -45 q^{-137} -57 q^{-138} - q^{-139} -29 q^{-140} +9 q^{-141} +12 q^{-142} +29 q^{-143} -7 q^{-144} -12 q^{-145} +5 q^{-146} -7 q^{-147} +5 q^{-150} -3 q^{-152} + q^{-153} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{49}-2 q^{48}+q^{47}+q^{46}-3 q^{45}+q^{44}+q^{43}+3 q^{42}+2 q^{41}-12 q^{40}+4 q^{39}+8 q^{38}-9 q^{37}+2 q^{36}+2 q^{35}+15 q^{34}+8 q^{33}-47 q^{32}-4 q^{31}+20 q^{30}-8 q^{29}+23 q^{28}+17 q^{27}+58 q^{26}+23 q^{25}-143 q^{24}-96 q^{23}-36 q^{22}-9 q^{21}+152 q^{20}+191 q^{19}+277 q^{18}+144 q^{17}-377 q^{16}-540 q^{15}-567 q^{14}-327 q^{13}+414 q^{12}+954 q^{11}+1430 q^{10}+1109 q^9-402 q^8-1739 q^7-2768 q^6-2573 q^5-380 q^4+2316 q^3+5012 q^2+5548 q+2388-2370 q^{-1} -7721 q^{-2} -10214 q^{-3} -6877 q^{-4} +623 q^{-5} +10484 q^{-6} +16861 q^{-7} +14375 q^{-8} +4244 q^{-9} -11540 q^{-10} -24697 q^{-11} -25884 q^{-12} -13868 q^{-13} +9592 q^{-14} +32412 q^{-15} +40381 q^{-16} +28983 q^{-17} -1848 q^{-18} -37526 q^{-19} -57242 q^{-20} -49960 q^{-21} -12323 q^{-22} +38102 q^{-23} +73166 q^{-24} +74995 q^{-25} +34149 q^{-26} -31469 q^{-27} -86279 q^{-28} -102321 q^{-29} -61840 q^{-30} +17521 q^{-31} +93437 q^{-32} +128109 q^{-33} +93677 q^{-34} +3851 q^{-35} -93581 q^{-36} -150173 q^{-37} -126202 q^{-38} -30131 q^{-39} +86429 q^{-40} +165930 q^{-41} +156317 q^{-42} +58993 q^{-43} -73228 q^{-44} -174769 q^{-45} -181702 q^{-46} -87249 q^{-47} +56185 q^{-48} +176989 q^{-49} +200831 q^{-50} +112483 q^{-51} -37527 q^{-52} -173938 q^{-53} -213557 q^{-54} -133252 q^{-55} +19431 q^{-56} +167410 q^{-57} +220610 q^{-58} +149010 q^{-59} -3390 q^{-60} -159021 q^{-61} -223169 q^{-62} -160099 q^{-63} -10179 q^{-64} +150009 q^{-65} +222740 q^{-66} +167515 q^{-67} +21258 q^{-68} -141101 q^{-69} -220251 q^{-70} -172291 q^{-71} -30665 q^{-72} +132140 q^{-73} +216507 q^{-74} +175603 q^{-75} +39315 q^{-76} -122795 q^{-77} -211642 q^{-78} -178042 q^{-79} -48156 q^{-80} +112108 q^{-81} +205276 q^{-82} +180019 q^{-83} +58062 q^{-84} -99274 q^{-85} -196848 q^{-86} -181260 q^{-87} -69252 q^{-88} +83491 q^{-89} +185321 q^{-90} +181081 q^{-91} +81626 q^{-92} -64344 q^{-93} -169860 q^{-94} -178470 q^{-95} -94293 q^{-96} +42096 q^{-97} +149911 q^{-98} +172010 q^{-99} +105726 q^{-100} -17597 q^{-101} -125246 q^{-102} -160638 q^{-103} -114329 q^{-104} -7281 q^{-105} +96865 q^{-106} +143639 q^{-107} +117833 q^{-108} +30299 q^{-109} -66111 q^{-110} -121194 q^{-111} -115123 q^{-112} -49018 q^{-113} +35809 q^{-114} +94658 q^{-115} +105401 q^{-116} +61050 q^{-117} -8534 q^{-118} -66080 q^{-119} -89559 q^{-120} -65447 q^{-121} -12985 q^{-122} +38731 q^{-123} +69413 q^{-124} +62059 q^{-125} +26964 q^{-126} -15169 q^{-127} -47756 q^{-128} -52635 q^{-129} -33026 q^{-130} -2300 q^{-131} +27660 q^{-132} +39557 q^{-133} +32200 q^{-134} +12759 q^{-135} -11470 q^{-136} -25676 q^{-137} -26572 q^{-138} -16839 q^{-139} +373 q^{-140} +13644 q^{-141} +18824 q^{-142} +16002 q^{-143} +5406 q^{-144} -4743 q^{-145} -11038 q^{-146} -12491 q^{-147} -7209 q^{-148} -564 q^{-149} +5046 q^{-150} +8191 q^{-151} +6327 q^{-152} +2770 q^{-153} -1113 q^{-154} -4371 q^{-155} -4446 q^{-156} -3090 q^{-157} -820 q^{-158} +1866 q^{-159} +2551 q^{-160} +2323 q^{-161} +1311 q^{-162} -374 q^{-163} -1079 q^{-164} -1465 q^{-165} -1226 q^{-166} -177 q^{-167} +368 q^{-168} +717 q^{-169} +757 q^{-170} +279 q^{-171} +71 q^{-172} -258 q^{-173} -484 q^{-174} -237 q^{-175} -99 q^{-176} +92 q^{-177} +198 q^{-178} +94 q^{-179} +115 q^{-180} +37 q^{-181} -100 q^{-182} -75 q^{-183} -62 q^{-184} -4 q^{-185} +42 q^{-186} -3 q^{-187} +29 q^{-188} +29 q^{-189} -9 q^{-190} -12 q^{-191} -20 q^{-192} -2 q^{-193} +12 q^{-194} -5 q^{-195} +7 q^{-197} -5 q^{-200} +3 q^{-202} - q^{-203} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 25]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 12, 6, 13], X[9, 17, 10, 16], X[13, 18, 14, 1], X[17, 14, 18, 15], X[15, 11, 16, 10], X[11, 6, 12, 7], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[9, 25]]
Out[3]=	GaussCode[-1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -8, 3, -5, 6, -7, 4, -6, 5]
In[4]:=	DTCode[Knot[9, 25]]
Out[4]=	DTCode[4, 8, 12, 2, 16, 6, 18, 10, 14]
In[5]:=	br = BR[Knot[9, 25]]
Out[5]=	BR[5, {-1, -1, 2, -1, -3, -2, -2, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 10}
In[7]:=	BraidIndex[Knot[9, 25]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[9, 25]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 25]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 3, {4, 7}, 1}
In[10]:=	alex = Alexander[Knot[9, 25]][t]
Out[10]=	3 12 2 -17 - -- + -- + 12 t - 3 t 2 t t
In[11]:=	Conway[Knot[9, 25]][z]
Out[11]=	4 1 - 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 25], Knot[11, NonAlternating, 134]}
In[13]:=	{KnotDet[Knot[9, 25]], KnotSignature[Knot[9, 25]]}
Out[13]=	{47, -2}
In[14]:=	Jones[Knot[9, 25]][q]
Out[14]=	-8 3 5 7 8 8 7 5 -2 - q + -- - -- + -- - -- + -- - -- + - + q 7 6 5 4 3 2 q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 25], Knot[11, NonAlternating, 25]}
In[16]:=	A2Invariant[Knot[9, 25]][q]
Out[16]=	-26 -24 2 -18 2 2 2 -6 -4 3 4 -q - q + --- + q + --- - --- - --- + q - q + -- + q 22 16 14 10 2 q q q q q
In[17]:=	HOMFLYPT[Knot[9, 25]][a, z]
Out[17]=	2 4 6 8 2 4 2 6 2 2 4 4 4 1 + a - 3 a + 3 a - a + z - 4 a z + 3 a z - a z - 2 a z
In[18]:=	Kauffman[Knot[9, 25]][a, z]
Out[18]=	2 4 6 8 3 5 7 9 2 1 - a - 3 a - 3 a - a - a z - a z + a z + a z - 2 z + 2 2 4 2 6 2 8 2 3 3 3 2 a z + 13 a z + 13 a z + 4 a z - 2 a z + 3 a z + 5 3 7 3 9 3 4 2 4 4 4 6 4 5 a z - 2 a z - 2 a z + z - 3 a z - 15 a z - 18 a z - 8 4 5 3 5 5 5 7 5 9 5 2 6 7 a z + 2 a z - 3 a z - 10 a z - 4 a z + a z + 3 a z + 4 6 6 6 8 6 3 7 5 7 7 7 4 8 6 a z + 6 a z + 3 a z + 3 a z + 6 a z + 3 a z + a z + 6 8 a z
In[19]:=	{Vassiliev[2][Knot[9, 25]], Vassiliev[3][Knot[9, 25]]}
Out[19]=	{0, -1}
In[20]:=	Kh[Knot[9, 25]][q, t]
Out[20]=	2 4 1 2 1 3 2 4 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 4 4 4 4 3 4 t 3 2 ----- + ----- + ----- + ----- + ---- + ---- + - + q t + q t 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[9, 25], 2][q]
Out[21]=	-23 3 10 12 7 30 20 25 52 19 -11 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - 22 20 19 18 17 16 15 14 13 q q q q q q q q q 45 65 12 56 62 3 50 43 4 31 19 4 --- + --- - --- - -- + -- - -- - -- + -- + -- - -- + -- + - + 5 q + 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 2 3 4 q - 2 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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