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]

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]=

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

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

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 25]]
Out[2]=	9
In[3]:=	PD[Knot[9, 25]]
Out[3]=	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[4]:=	GaussCode[Knot[9, 25]]
Out[4]=	GaussCode[-1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -8, 3, -5, 6, -7, 4, -6, 5]
In[5]:=	BR[Knot[9, 25]]
Out[5]=	BR[5, {-1, -1, 2, -1, -3, -2, -2, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[9, 25]][t]
Out[6]=	3 12 2 -17 - -- + -- + 12 t - 3 t 2 t t
In[7]:=	Conway[Knot[9, 25]][z]
Out[7]=	4 1 - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 25], Knot[11, NonAlternating, 134]}
In[9]:=	{KnotDet[Knot[9, 25]], KnotSignature[Knot[9, 25]]}
Out[9]=	{47, -2}
In[10]:=	J=Jones[Knot[9, 25]][q]
Out[10]=	-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[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 25], Knot[11, NonAlternating, 25]}
In[12]:=	A2Invariant[Knot[9, 25]][q]
Out[12]=	-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[13]:=	Kauffman[Knot[9, 25]][a, z]
Out[13]=	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[14]:=	{Vassiliev[2][Knot[9, 25]], Vassiliev[3][Knot[9, 25]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[9, 25]][q, t]
Out[15]=	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