9 24

9_23

9_25

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 24's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 24 at Knotilus!

Knot presentations

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

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{12, 4}, {3, 10}, {8, 11}, {10, 12}, {9, 5}, {4, 8}, {6, 9}, {5, 7}, {2, 6}, {1, 3}, {11, 2}, {7, 1}]

[edit Notes on presentations of 9 24]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 24"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X5,14,6,15 X9,17,10,16 X11,1,12,18 X17,11,18,10 X15,13,16,12 X13,6,14,7 X7283

In[5]:=

GaussCode[K]

Out[5]=

-1, 9, -2, 1, -3, 8, -9, 2, -4, 6, -5, 7, -8, 3, -7, 4, -6, 5

In[6]:=

DTCode[K]

Out[6]=

4 8 14 2 16 18 6 12 10

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[3,21,2+]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(4,\{-1,-1,2,-1,-3,2,2,2,-3\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ 4, 9, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{12, 4}, {3, 10}, {8, 11}, {10, 12}, {9, 5}, {4, 8}, {6, 9}, {5, 7}, {2, 6}, {1, 3}, {11, 2}, {7, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 10.8337 A-Polynomial See Data:9 24/A-polynomial

[edit Notes for 9 24's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $1$ Rasmussen s-Invariant 0

[edit Notes for 9 24's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-10t+13-10t^{-1}+5t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 45, 0 }
Jones polynomial	$q^{4}-3q^{3}+5q^{2}-7q+8-7q^{-1}+7q^{-2}-4q^{-3}+2q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-4z^{4}-a^{4}z^{2}+6a^{2}z^{2}+2z^{2}a^{-2}-6z^{2}-2a^{4}+5a^{2}+a^{-2}-3$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+2a^{3}z^{7}+5az^{7}+3z^{7}a^{-1}+2a^{4}z^{6}+3a^{2}z^{6}+4z^{6}a^{-2}+5z^{6}+a^{5}z^{5}-2a^{3}z^{5}-7az^{5}-z^{5}a^{-1}+3z^{5}a^{-3}-5a^{4}z^{4}-10a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}-11z^{4}-3a^{5}z^{3}-3a^{3}z^{3}+az^{3}-3z^{3}a^{-1}-4z^{3}a^{-3}+4a^{4}z^{2}+10a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+9z^{2}+2a^{5}z+3a^{3}z+2az+2za^{-1}+za^{-3}-2a^{4}-5a^{2}-a^{-2}-3$
The A2 invariant	$-q^{16}-q^{14}-q^{10}+3q^{8}+2q^{6}+q^{4}+2q^{2}-2+q^{-2}-2q^{-4}+q^{-8}-q^{-10}+q^{-12}$
The G2 invariant	$q^{80}-q^{78}+3q^{76}-4q^{74}+3q^{72}-3q^{70}-3q^{68}+9q^{66}-16q^{64}+19q^{62}-19q^{60}+8q^{58}+7q^{56}-26q^{54}+41q^{52}-47q^{50}+34q^{48}-11q^{46}-23q^{44}+45q^{42}-53q^{40}+46q^{38}-16q^{36}-11q^{34}+34q^{32}-36q^{30}+22q^{28}+10q^{26}-32q^{24}+44q^{22}-27q^{20}+2q^{18}+37q^{16}-60q^{14}+71q^{12}-55q^{10}+21q^{8}+20q^{6}-60q^{4}+77q^{2}-69+40q^{-2}-4q^{-4}-32q^{-6}+46q^{-8}-43q^{-10}+18q^{-12}+8q^{-14}-31q^{-16}+33q^{-18}-15q^{-20}-14q^{-22}+41q^{-24}-50q^{-26}+44q^{-28}-20q^{-30}-11q^{-32}+35q^{-34}-48q^{-36}+47q^{-38}-29q^{-40}+9q^{-42}+9q^{-44}-21q^{-46}+24q^{-48}-19q^{-50}+13q^{-52}-4q^{-54}-2q^{-56}+4q^{-58}-6q^{-60}+4q^{-62}-2q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_24.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+q^{9}-2q^{7}+3q^{5}+q+q^{-1}-2q^{-3}+2q^{-5}-2q^{-7}+q^{-9}$
2	$q^{32}-q^{30}-q^{28}+4q^{26}-3q^{24}-6q^{22}+9q^{20}-q^{18}-12q^{16}+11q^{14}+5q^{12}-12q^{10}+5q^{8}+7q^{6}-5q^{4}-2q^{2}+5+5q^{-2}-10q^{-4}+12q^{-8}-11q^{-10}-4q^{-12}+13q^{-14}-5q^{-16}-5q^{-18}+6q^{-20}-q^{-22}-2q^{-24}+q^{-26}$
3	$-q^{63}+q^{61}+q^{59}-q^{57}-3q^{55}+3q^{53}+7q^{51}-3q^{49}-13q^{47}+q^{45}+21q^{43}+5q^{41}-30q^{39}-17q^{37}+34q^{35}+30q^{33}-30q^{31}-48q^{29}+26q^{27}+54q^{25}-11q^{23}-59q^{21}-q^{19}+55q^{17}+12q^{15}-46q^{13}-18q^{11}+34q^{9}+25q^{7}-16q^{5}-30q^{3}+3q+32q^{-1}+17q^{-3}-36q^{-5}-32q^{-7}+32q^{-9}+49q^{-11}-27q^{-13}-56q^{-15}+16q^{-17}+57q^{-19}-3q^{-21}-53q^{-23}-7q^{-25}+41q^{-27}+13q^{-29}-26q^{-31}-14q^{-33}+14q^{-35}+11q^{-37}-8q^{-39}-5q^{-41}+3q^{-43}+3q^{-45}-q^{-47}-2q^{-49}+q^{-51}$
4	$q^{104}-q^{102}-q^{100}+q^{98}+3q^{94}-5q^{92}-5q^{90}+5q^{88}+5q^{86}+13q^{84}-13q^{82}-24q^{80}+18q^{76}+51q^{74}-7q^{72}-62q^{70}-48q^{68}+6q^{66}+121q^{64}+61q^{62}-67q^{60}-138q^{58}-92q^{56}+148q^{54}+182q^{52}+34q^{50}-177q^{48}-244q^{46}+62q^{44}+244q^{42}+183q^{40}-104q^{38}-320q^{36}-78q^{34}+190q^{32}+264q^{30}+10q^{28}-275q^{26}-155q^{24}+89q^{22}+236q^{20}+81q^{18}-164q^{16}-163q^{14}-5q^{12}+160q^{10}+121q^{8}-41q^{6}-149q^{4}-95q^{2}+70+162q^{-2}+100q^{-4}-130q^{-6}-197q^{-8}-46q^{-10}+180q^{-12}+245q^{-14}-61q^{-16}-257q^{-18}-180q^{-20}+124q^{-22}+327q^{-24}+56q^{-26}-203q^{-28}-251q^{-30}-3q^{-32}+269q^{-34}+135q^{-36}-65q^{-38}-202q^{-40}-92q^{-42}+129q^{-44}+108q^{-46}+32q^{-48}-86q^{-50}-80q^{-52}+28q^{-54}+36q^{-56}+36q^{-58}-18q^{-60}-31q^{-62}+6q^{-64}+3q^{-66}+12q^{-68}-3q^{-70}-8q^{-72}+3q^{-74}+3q^{-78}-q^{-80}-2q^{-82}+q^{-84}$
5	$-q^{155}+q^{153}+q^{151}-q^{149}-q^{143}+3q^{141}+4q^{139}-5q^{137}-8q^{135}-3q^{133}+3q^{131}+15q^{129}+18q^{127}-4q^{125}-33q^{123}-37q^{121}-7q^{119}+45q^{117}+80q^{115}+44q^{113}-51q^{111}-137q^{109}-117q^{107}+22q^{105}+191q^{103}+236q^{101}+75q^{99}-208q^{97}-385q^{95}-252q^{93}+141q^{91}+507q^{89}+504q^{87}+49q^{85}-542q^{83}-782q^{81}-365q^{79}+452q^{77}+991q^{75}+743q^{73}-182q^{71}-1079q^{69}-1137q^{67}-171q^{65}+1004q^{63}+1392q^{61}+604q^{59}-770q^{57}-1536q^{55}-963q^{53}+466q^{51}+1481q^{49}+1212q^{47}-122q^{45}-1326q^{43}-1310q^{41}-155q^{39}+1078q^{37}+1280q^{35}+345q^{33}-813q^{31}-1161q^{29}-460q^{27}+570q^{25}+999q^{23}+517q^{21}-356q^{19}-836q^{17}-557q^{15}+171q^{13}+691q^{11}+615q^{9}+34q^{7}-577q^{5}-700q^{3}-242q+443q^{-1}+820q^{-3}+521q^{-5}-300q^{-7}-942q^{-9}-808q^{-11}+70q^{-13}+1011q^{-15}+1140q^{-17}+203q^{-19}-1002q^{-21}-1395q^{-23}-550q^{-25}+857q^{-27}+1563q^{-29}+895q^{-31}-596q^{-33}-1567q^{-35}-1165q^{-37}+236q^{-39}+1391q^{-41}+1310q^{-43}+137q^{-45}-1074q^{-47}-1286q^{-49}-423q^{-51}+677q^{-53}+1100q^{-55}+588q^{-57}-303q^{-59}-818q^{-61}-603q^{-63}+32q^{-65}+508q^{-67}+498q^{-69}+118q^{-71}-248q^{-73}-349q^{-75}-160q^{-77}+94q^{-79}+196q^{-81}+126q^{-83}-7q^{-85}-90q^{-87}-85q^{-89}-12q^{-91}+40q^{-93}+36q^{-95}+11q^{-97}-11q^{-99}-15q^{-101}-8q^{-103}+6q^{-105}+8q^{-107}-2q^{-109}-3q^{-111}+3q^{-119}-q^{-121}-2q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}-q^{14}-q^{10}+3q^{8}+2q^{6}+q^{4}+2q^{2}-2+q^{-2}-2q^{-4}+q^{-8}-q^{-10}+q^{-12}$
1,1	$q^{44}-2q^{42}+6q^{40}-12q^{38}+25q^{36}-42q^{34}+66q^{32}-100q^{30}+130q^{28}-168q^{26}+186q^{24}-196q^{22}+177q^{20}-132q^{18}+72q^{16}+26q^{14}-114q^{12}+216q^{10}-294q^{8}+352q^{6}-385q^{4}+376q^{2}-338+268q^{-2}-177q^{-4}+80q^{-6}+16q^{-8}-96q^{-10}+156q^{-12}-186q^{-14}+192q^{-16}-178q^{-18}+152q^{-20}-120q^{-22}+86q^{-24}-58q^{-26}+36q^{-28}-20q^{-30}+10q^{-32}-4q^{-34}+q^{-36}$
2,0	$q^{42}+q^{40}-q^{36}+q^{34}+q^{32}-4q^{30}-6q^{28}+q^{24}-4q^{22}+8q^{18}+7q^{16}-3q^{14}+2q^{12}+5q^{10}-3q^{8}-3q^{6}+3q^{4}-4+2q^{-2}+4q^{-4}-4q^{-6}-3q^{-8}+6q^{-10}+2q^{-12}-6q^{-14}+q^{-16}+5q^{-18}-q^{-20}-3q^{-22}+2q^{-26}-q^{-28}-q^{-30}+q^{-32}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-q^{32}+q^{30}+2q^{28}-5q^{26}+2q^{22}-12q^{20}+q^{18}+8q^{16}-8q^{14}+8q^{12}+13q^{10}-q^{8}+q^{6}+4q^{4}-2q^{2}-6-5q^{-2}+5q^{-4}-3q^{-6}-7q^{-8}+12q^{-10}-8q^{-14}+9q^{-16}-6q^{-20}+4q^{-22}-2q^{-26}+q^{-28}$
1,0,0	$-q^{21}-q^{19}-2q^{17}-q^{13}+4q^{11}+2q^{9}+4q^{7}+q^{5}+q^{3}-q-2q^{-1}-2q^{-5}+q^{-7}-q^{-9}+2q^{-11}-q^{-13}+q^{-15}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{44}+q^{42}+q^{40}+q^{38}+q^{36}-2q^{34}-5q^{32}-4q^{30}-4q^{28}-11q^{26}-7q^{24}+6q^{22}+5q^{20}+2q^{18}+14q^{16}+20q^{14}+6q^{12}+5q^{8}-q^{6}-16q^{4}-6q^{2}+1-9q^{-2}-5q^{-4}+10q^{-6}+2q^{-8}-5q^{-10}+6q^{-12}+8q^{-14}-3q^{-16}-5q^{-18}+5q^{-20}+2q^{-22}-5q^{-24}-q^{-26}+3q^{-28}-q^{-30}-q^{-32}+q^{-34}$
1,0,0,0	$-q^{26}-q^{24}-2q^{22}-2q^{20}-q^{16}+4q^{14}+3q^{12}+4q^{10}+4q^{8}+q^{6}+q^{4}-2q^{2}-1-3q^{-2}-2q^{-6}+q^{-8}+2q^{-14}-q^{-16}+q^{-18}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+q^{32}-3q^{30}+4q^{28}-7q^{26}+8q^{24}-10q^{22}+10q^{20}-9q^{18}+8q^{16}-2q^{14}+9q^{10}-11q^{8}+17q^{6}-18q^{4}+20q^{2}-20+15q^{-2}-11q^{-4}+5q^{-6}-q^{-8}-4q^{-10}+8q^{-12}-10q^{-14}+11q^{-16}-10q^{-18}+8q^{-20}-6q^{-22}+4q^{-24}-2q^{-26}+q^{-28}$
1,0	$q^{56}-q^{52}-q^{50}+2q^{48}+3q^{46}-q^{44}-6q^{42}-3q^{40}+5q^{38}+6q^{36}-5q^{34}-12q^{32}-3q^{30}+10q^{28}+8q^{26}-7q^{24}-8q^{22}+4q^{20}+13q^{18}+3q^{16}-5q^{14}-q^{12}+8q^{10}+4q^{8}-5q^{6}-6q^{4}+3q^{2}+6-4q^{-2}-9q^{-4}+8q^{-8}+q^{-10}-9q^{-12}-5q^{-14}+9q^{-16}+10q^{-18}-4q^{-20}-11q^{-22}-q^{-24}+10q^{-26}+6q^{-28}-5q^{-30}-7q^{-32}+5q^{-36}+2q^{-38}-2q^{-40}-2q^{-42}+q^{-46}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-q^{44}+2q^{42}-2q^{40}+4q^{38}-6q^{36}+4q^{34}-9q^{32}+5q^{30}-12q^{28}+4q^{26}-8q^{24}+7q^{22}-q^{20}+4q^{18}+9q^{16}+3q^{14}+15q^{12}-8q^{10}+14q^{8}-13q^{6}+14q^{4}-19q^{2}+9-17q^{-2}+9q^{-4}-8q^{-6}+3q^{-8}-3q^{-10}+q^{-12}+7q^{-14}-4q^{-16}+7q^{-18}-8q^{-20}+10q^{-22}-7q^{-24}+6q^{-26}-7q^{-28}+5q^{-30}-3q^{-32}+2q^{-34}-2q^{-36}+q^{-38}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-q^{78}+3q^{76}-4q^{74}+3q^{72}-3q^{70}-3q^{68}+9q^{66}-16q^{64}+19q^{62}-19q^{60}+8q^{58}+7q^{56}-26q^{54}+41q^{52}-47q^{50}+34q^{48}-11q^{46}-23q^{44}+45q^{42}-53q^{40}+46q^{38}-16q^{36}-11q^{34}+34q^{32}-36q^{30}+22q^{28}+10q^{26}-32q^{24}+44q^{22}-27q^{20}+2q^{18}+37q^{16}-60q^{14}+71q^{12}-55q^{10}+21q^{8}+20q^{6}-60q^{4}+77q^{2}-69+40q^{-2}-4q^{-4}-32q^{-6}+46q^{-8}-43q^{-10}+18q^{-12}+8q^{-14}-31q^{-16}+33q^{-18}-15q^{-20}-14q^{-22}+41q^{-24}-50q^{-26}+44q^{-28}-20q^{-30}-11q^{-32}+35q^{-34}-48q^{-36}+47q^{-38}-29q^{-40}+9q^{-42}+9q^{-44}-21q^{-46}+24q^{-48}-19q^{-50}+13q^{-52}-4q^{-54}-2q^{-56}+4q^{-58}-6q^{-60}+4q^{-62}-2q^{-64}+q^{-66}$

.

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 24"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+5t^{2}-10t+13-10t^{-1}+5t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}-z^{4}+z^{2}+1$

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]=

{ 45, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-4z^{4}-a^{4}z^{2}+6a^{2}z^{2}+2z^{2}a^{-2}-6z^{2}-2a^{4}+5a^{2}+a^{-2}-3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{8}+z^{8}+2a^{3}z^{7}+5az^{7}+3z^{7}a^{-1}+2a^{4}z^{6}+3a^{2}z^{6}+4z^{6}a^{-2}+5z^{6}+a^{5}z^{5}-2a^{3}z^{5}-7az^{5}-z^{5}a^{-1}+3z^{5}a^{-3}-5a^{4}z^{4}-10a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}-11z^{4}-3a^{5}z^{3}-3a^{3}z^{3}+az^{3}-3z^{3}a^{-1}-4z^{3}a^{-3}+4a^{4}z^{2}+10a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+9z^{2}+2a^{5}z+3a^{3}z+2az+2za^{-1}+za^{-3}-2a^{4}-5a^{2}-a^{-2}-3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_18, K11n85, K11n164,}

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

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 24"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]=

{ $-t^{3}+5t^{2}-10t+13-10t^{-1}+5t^{-2}-t^{-3}$, $q^{4}-3q^{3}+5q^{2}-7q+8-7q^{-1}+7q^{-2}-4q^{-3}+2q^{-4}-q^{-5}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{8_18, K11n85, K11n164,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(1, -2)

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 $4$ $-16$ $8$ ${\frac {110}{3}}$ ${\frac {34}{3}}$ $-64$ $-{\frac {352}{3}}$ ${\frac {32}{3}}$ $-48$ ${\frac {32}{3}}$ $128$ ${\frac {440}{3}}$ ${\frac {136}{3}}$ ${\frac {13471}{30}}$ $-{\frac {782}{15}}$ ${\frac {10862}{45}}$ ${\frac {545}{18}}$ ${\frac {991}{30}}$

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=$0 is the signature of 9 24. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 2   -2 5               3 1   2 3             4 2     -2 1           4 3       1 -1         4 5         1 -3       3 3           0 -5     1 4             3 -7   1 3               -2 -9   1                 1 -11 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{12}-3q^{11}+q^{10}+8q^{9}-14q^{8}+q^{7}+26q^{6}-31q^{5}-6q^{4}+49q^{3}-43q^{2}-16q+64-43q^{-1}-23q^{-2}+61q^{-3}-31q^{-4}-25q^{-5}+44q^{-6}-14q^{-7}-19q^{-8}+21q^{-9}-3q^{-10}-9q^{-11}+6q^{-12}-2q^{-14}+q^{-15}$
3	$q^{24}-3q^{23}+q^{22}+4q^{21}+q^{20}-11q^{19}-2q^{18}+23q^{17}+4q^{16}-39q^{15}-14q^{14}+62q^{13}+32q^{12}-87q^{11}-60q^{10}+112q^{9}+92q^{8}-128q^{7}-132q^{6}+141q^{5}+168q^{4}-145q^{3}-196q^{2}+137q+221-130q^{-1}-225q^{-2}+104q^{-3}+235q^{-4}-89q^{-5}-216q^{-6}+52q^{-7}+207q^{-8}-31q^{-9}-173q^{-10}-4q^{-11}+149q^{-12}+17q^{-13}-108q^{-14}-32q^{-15}+75q^{-16}+35q^{-17}-48q^{-18}-28q^{-19}+24q^{-20}+22q^{-21}-13q^{-22}-12q^{-23}+4q^{-24}+8q^{-25}-3q^{-26}-2q^{-27}+2q^{-29}-q^{-30}$
4	$q^{40}-3q^{39}+q^{38}+4q^{37}-3q^{36}+4q^{35}-14q^{34}+6q^{33}+19q^{32}-12q^{31}+7q^{30}-51q^{29}+19q^{28}+73q^{27}-12q^{26}-q^{25}-159q^{24}+13q^{23}+191q^{22}+64q^{21}+20q^{20}-380q^{19}-97q^{18}+328q^{17}+264q^{16}+154q^{15}-652q^{14}-345q^{13}+376q^{12}+523q^{11}+425q^{10}-855q^{9}-649q^{8}+299q^{7}+719q^{6}+731q^{5}-920q^{4}-875q^{3}+148q^{2}+786q+961-858q^{-1}-967q^{-2}-17q^{-3}+732q^{-4}+1069q^{-5}-696q^{-6}-928q^{-7}-182q^{-8}+574q^{-9}+1068q^{-10}-451q^{-11}-773q^{-12}-329q^{-13}+330q^{-14}+948q^{-15}-166q^{-16}-519q^{-17}-403q^{-18}+62q^{-19}+706q^{-20}+50q^{-21}-232q^{-22}-342q^{-23}-120q^{-24}+400q^{-25}+117q^{-26}-21q^{-27}-194q^{-28}-154q^{-29}+160q^{-30}+71q^{-31}+50q^{-32}-66q^{-33}-94q^{-34}+45q^{-35}+17q^{-36}+36q^{-37}-11q^{-38}-36q^{-39}+12q^{-40}-q^{-41}+12q^{-42}-10q^{-44}+4q^{-45}-q^{-46}+2q^{-47}-2q^{-49}+q^{-50}$
5	$q^{60}-3q^{59}+q^{58}+4q^{57}-3q^{56}+q^{54}-6q^{53}+2q^{52}+14q^{51}-5q^{50}-14q^{49}-6q^{48}-2q^{47}+24q^{46}+39q^{45}-q^{44}-66q^{43}-79q^{42}-7q^{41}+107q^{40}+172q^{39}+69q^{38}-168q^{37}-333q^{36}-196q^{35}+208q^{34}+538q^{33}+449q^{32}-158q^{31}-809q^{30}-831q^{29}-7q^{28}+1053q^{27}+1340q^{26}+354q^{25}-1232q^{24}-1931q^{23}-870q^{22}+1265q^{21}+2551q^{20}+1527q^{19}-1151q^{18}-3086q^{17}-2271q^{16}+863q^{15}+3522q^{14}+3018q^{13}-483q^{12}-3792q^{11}-3678q^{10}+18q^{9}+3915q^{8}+4223q^{7}+454q^{6}-3921q^{5}-4619q^{4}-860q^{3}+3781q^{2}+4865q+1275-3622q^{-1}-4996q^{-2}-1545q^{-3}+3323q^{-4}+4988q^{-5}+1886q^{-6}-3041q^{-7}-4920q^{-8}-2065q^{-9}+2595q^{-10}+4709q^{-11}+2366q^{-12}-2168q^{-13}-4438q^{-14}-2494q^{-15}+1565q^{-16}+4008q^{-17}+2714q^{-18}-1010q^{-19}-3503q^{-20}-2696q^{-21}+332q^{-22}+2853q^{-23}+2698q^{-24}+194q^{-25}-2169q^{-26}-2427q^{-27}-683q^{-28}+1424q^{-29}+2125q^{-30}+960q^{-31}-795q^{-32}-1639q^{-33}-1071q^{-34}+249q^{-35}+1159q^{-36}+1018q^{-37}+102q^{-38}-714q^{-39}-823q^{-40}-290q^{-41}+342q^{-42}+601q^{-43}+342q^{-44}-123q^{-45}-368q^{-46}-287q^{-47}-24q^{-48}+208q^{-49}+209q^{-50}+54q^{-51}-85q^{-52}-126q^{-53}-69q^{-54}+39q^{-55}+70q^{-56}+34q^{-57}+q^{-58}-31q^{-59}-33q^{-60}+4q^{-61}+18q^{-62}+4q^{-63}+5q^{-64}-2q^{-65}-11q^{-66}+q^{-67}+6q^{-68}-2q^{-69}+q^{-71}-2q^{-72}+2q^{-74}-q^{-75}$
6	$q^{84}-3q^{83}+q^{82}+4q^{81}-3q^{80}-3q^{78}+9q^{77}-10q^{76}-3q^{75}+21q^{74}-15q^{73}-8q^{72}-11q^{71}+36q^{70}-9q^{69}-2q^{68}+54q^{67}-59q^{66}-65q^{65}-60q^{64}+110q^{63}+45q^{62}+83q^{61}+181q^{60}-163q^{59}-293q^{58}-343q^{57}+118q^{56}+196q^{55}+480q^{54}+747q^{53}-88q^{52}-761q^{51}-1276q^{50}-510q^{49}+26q^{48}+1292q^{47}+2386q^{46}+1063q^{45}-834q^{44}-2971q^{43}-2645q^{42}-1728q^{41}+1621q^{40}+5148q^{39}+4384q^{38}+1092q^{37}-4217q^{36}-6238q^{35}-6260q^{34}-381q^{33}+7486q^{32}+9593q^{31}+6158q^{30}-2964q^{29}-9446q^{28}-12930q^{27}-5661q^{26}+7256q^{25}+14508q^{24}+13314q^{23}+1504q^{22}-10137q^{21}-19221q^{20}-12791q^{19}+3969q^{18}+16960q^{17}+19867q^{16}+7571q^{15}-8007q^{14}-22977q^{13}-19097q^{12}-756q^{11}+16666q^{10}+23858q^{9}+12843q^{8}-4544q^{7}-23940q^{6}-22954q^{5}-4949q^{4}+14828q^{3}+25188q^{2}+16161q-1272-23090q^{-1}-24468q^{-2}-7896q^{-3}+12518q^{-4}+24741q^{-5}+17851q^{-6}+1461q^{-7}-21188q^{-8}-24457q^{-9}-10067q^{-10}+9789q^{-11}+23057q^{-12}+18658q^{-13}+4229q^{-14}-18113q^{-15}-23290q^{-16}-12085q^{-17}+6069q^{-18}+19884q^{-19}+18710q^{-20}+7474q^{-21}-13340q^{-22}-20537q^{-23}-13752q^{-24}+1184q^{-25}+14726q^{-26}+17254q^{-27}+10578q^{-28}-6988q^{-29}-15602q^{-30}-13882q^{-31}-3759q^{-32}+7912q^{-33}+13444q^{-34}+11899q^{-35}-611q^{-36}-8900q^{-37}-11356q^{-38}-6693q^{-39}+1289q^{-40}+7705q^{-41}+10190q^{-42}+3436q^{-43}-2472q^{-44}-6675q^{-45}-6341q^{-46}-2739q^{-47}+2197q^{-48}+6189q^{-49}+4001q^{-50}+1342q^{-51}-2113q^{-52}-3676q^{-53}-3349q^{-54}-896q^{-55}+2304q^{-56}+2309q^{-57}+2036q^{-58}+348q^{-59}-1054q^{-60}-1998q^{-61}-1390q^{-62}+255q^{-63}+595q^{-64}+1151q^{-65}+750q^{-66}+157q^{-67}-683q^{-68}-757q^{-69}-188q^{-70}-111q^{-71}+337q^{-72}+378q^{-73}+295q^{-74}-124q^{-75}-238q^{-76}-78q^{-77}-154q^{-78}+34q^{-79}+98q^{-80}+145q^{-81}-10q^{-82}-54q^{-83}+5q^{-84}-62q^{-85}-11q^{-86}+11q^{-87}+48q^{-88}-4q^{-89}-15q^{-90}+14q^{-91}-15q^{-92}-4q^{-93}-2q^{-94}+13q^{-95}-2q^{-96}-7q^{-97}+6q^{-98}-2q^{-99}-q^{-101}+2q^{-102}-2q^{-104}+q^{-105}$
7	$q^{112}-3q^{111}+q^{110}+4q^{109}-3q^{108}-3q^{106}+5q^{105}+5q^{104}-15q^{103}+4q^{102}+11q^{101}-9q^{100}-2q^{99}-12q^{98}+19q^{97}+37q^{96}-32q^{95}-2q^{94}-48q^{92}-14q^{91}-40q^{90}+72q^{89}+168q^{88}+24q^{87}+16q^{86}-91q^{85}-259q^{84}-181q^{83}-188q^{82}+155q^{81}+590q^{80}+477q^{79}+407q^{78}-137q^{77}-875q^{76}-1031q^{75}-1120q^{74}-203q^{73}+1281q^{72}+1989q^{71}+2345q^{70}+1079q^{69}-1362q^{68}-3110q^{67}-4428q^{66}-3118q^{65}+618q^{64}+4273q^{63}+7457q^{62}+6604q^{61}+1553q^{60}-4636q^{59}-10927q^{58}-11983q^{57}-6169q^{56}+3303q^{55}+14408q^{54}+19031q^{53}+13506q^{52}+831q^{51}-16442q^{50}-26995q^{49}-23947q^{48}-8676q^{47}+15902q^{46}+34749q^{45}+36681q^{44}+20480q^{43}-11524q^{42}-40702q^{41}-50508q^{40}-35873q^{39}+2765q^{38}+43450q^{37}+63855q^{36}+53648q^{35}+10102q^{34}-42189q^{33}-75024q^{32}-71983q^{31}-26135q^{30}+36687q^{29}+82781q^{28}+89294q^{27}+43774q^{26}-27768q^{25}-86691q^{24}-103909q^{23}-61115q^{22}+16431q^{21}+86756q^{20}+115077q^{19}+76927q^{18}-4269q^{17}-83961q^{16}-122499q^{15}-89954q^{14}-7438q^{13}+79102q^{12}+126507q^{11}+99926q^{10}+17883q^{9}-73296q^{8}-127958q^{7}-106970q^{6}-26389q^{5}+67443q^{4}+127324q^{3}+111375q^{2}+33314q-61653-125691q^{-1}-114184q^{-2}-38579q^{-3}+56492q^{-4}+123025q^{-5}+115435q^{-6}+43195q^{-7}-51048q^{-8}-119959q^{-9}-116286q^{-10}-47264q^{-11}+45652q^{-12}+115863q^{-13}+116152q^{-14}+51725q^{-15}-38925q^{-16}-110840q^{-17}-115841q^{-18}-56394q^{-19}+31289q^{-20}+103934q^{-21}+114155q^{-22}+61680q^{-23}-21528q^{-24}-95151q^{-25}-111441q^{-26}-66761q^{-27}+10615q^{-28}+83603q^{-29}+106145q^{-30}+71494q^{-31}+1936q^{-32}-69811q^{-33}-98629q^{-34}-74307q^{-35}-14326q^{-36}+53509q^{-37}+87585q^{-38}+74888q^{-39}+26325q^{-40}-36303q^{-41}-74054q^{-42}-71793q^{-43}-35554q^{-44}+18850q^{-45}+57750q^{-46}+65265q^{-47}+41743q^{-48}-3220q^{-49}-40863q^{-50}-55196q^{-51}-43277q^{-52}-9314q^{-53}+24128q^{-54}+42753q^{-55}+40895q^{-56}+17656q^{-57}-9884q^{-58}-29576q^{-59}-34702q^{-60}-21292q^{-61}-1049q^{-62}+17056q^{-63}+26466q^{-64}+21009q^{-65}+7799q^{-66}-6956q^{-67}-17714q^{-68}-17580q^{-69}-10597q^{-70}-194q^{-71}+9806q^{-72}+12797q^{-73}+10460q^{-74}+4065q^{-75}-3942q^{-76}-7862q^{-77}-8324q^{-78}-5364q^{-79}+153q^{-80}+3845q^{-81}+5689q^{-82}+4915q^{-83}+1542q^{-84}-1117q^{-85}-3181q^{-86}-3682q^{-87}-1983q^{-88}-293q^{-89}+1431q^{-90}+2297q^{-91}+1615q^{-92}+851q^{-93}-342q^{-94}-1288q^{-95}-1099q^{-96}-782q^{-97}-83q^{-98}+543q^{-99}+559q^{-100}+625q^{-101}+267q^{-102}-237q^{-103}-296q^{-104}-346q^{-105}-172q^{-106}+44q^{-107}+50q^{-108}+208q^{-109}+172q^{-110}-16q^{-111}-40q^{-112}-97q^{-113}-48q^{-114}+3q^{-115}-38q^{-116}+38q^{-117}+60q^{-118}+4q^{-119}-q^{-120}-27q^{-121}-5q^{-122}+13q^{-123}-21q^{-124}+q^{-125}+14q^{-126}+2q^{-127}+2q^{-128}-9q^{-129}+8q^{-131}-5q^{-132}-2q^{-133}+2q^{-134}+q^{-136}-2q^{-137}+2q^{-139}-q^{-140}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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