10 45

10_44

10_46

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10 45 Quick Notes

10 45 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X12,6,13,5 X10,3,11,4 X2,11,3,12 X20,14,1,13 X14,7,15,8 X6,19,7,20 X18,15,19,16 X16,10,17,9 X8,18,9,17
Gauss code	1, -4, 3, -1, 2, -7, 6, -10, 9, -3, 4, -2, 5, -6, 8, -9, 10, -8, 7, -5
Dowker-Thistlethwaite code	4 10 12 14 16 2 20 18 8 6
Conway Notation	[21111112]

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 13.7161 A-Polynomial See Data:10 45/A-polynomial

[edit Notes for 10 45's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ 3 }[/math] Rasmussen s-Invariant 0

[edit Notes for 10 45's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+7 t^2-21 t+31-21 t^{-1} +7 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6+z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 89, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+4 q^4-7 q^3+11 q^2-14 q+15-14 q^{-1} +11 q^{-2} -7 q^{-3} +4 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+2 a^2 z^4+2 z^4 a^{-2} -3 z^4-a^4 z^2+3 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} -6 z^2+2 a^2+2 a^{-2} -3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a z^9+z^9 a^{-1} +4 a^2 z^8+4 z^8 a^{-2} +8 z^8+6 a^3 z^7+14 a z^7+14 z^7 a^{-1} +6 z^7 a^{-3} +4 a^4 z^6+3 a^2 z^6+3 z^6 a^{-2} +4 z^6 a^{-4} -2 z^6+a^5 z^5-10 a^3 z^5-31 a z^5-31 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -7 a^4 z^4-17 a^2 z^4-17 z^4 a^{-2} -7 z^4 a^{-4} -20 z^4-a^5 z^3+5 a^3 z^3+21 a z^3+21 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +3 a^4 z^2+12 a^2 z^2+12 z^2 a^{-2} +3 z^2 a^{-4} +18 z^2-a^3 z-5 a z-5 z a^{-1} -z a^{-3} -2 a^2-2 a^{-2} -3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{14}+2 q^{12}-2 q^{10}+3 q^8-2 q^4+2 q^2-3+2 q^{-2} -2 q^{-4} +3 q^{-8} -2 q^{-10} +2 q^{-12} + q^{-14} - q^{-16} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+14 q^{72}-12 q^{70}-q^{68}+26 q^{66}-51 q^{64}+77 q^{62}-84 q^{60}+57 q^{58}-82 q^{54}+162 q^{52}-205 q^{50}+193 q^{48}-112 q^{46}-20 q^{44}+163 q^{42}-263 q^{40}+285 q^{38}-209 q^{36}+66 q^{34}+90 q^{32}-201 q^{30}+222 q^{28}-143 q^{26}+10 q^{24}+123 q^{22}-188 q^{20}+146 q^{18}-16 q^{16}-156 q^{14}+293 q^{12}-328 q^{10}+240 q^8-49 q^6-182 q^4+363 q^2-433+363 q^{-2} -182 q^{-4} -49 q^{-6} +240 q^{-8} -328 q^{-10} +293 q^{-12} -156 q^{-14} -16 q^{-16} +146 q^{-18} -188 q^{-20} +123 q^{-22} +10 q^{-24} -143 q^{-26} +222 q^{-28} -201 q^{-30} +90 q^{-32} +66 q^{-34} -209 q^{-36} +285 q^{-38} -263 q^{-40} +163 q^{-42} -20 q^{-44} -112 q^{-46} +193 q^{-48} -205 q^{-50} +162 q^{-52} -82 q^{-54} +57 q^{-58} -84 q^{-60} +77 q^{-62} -51 q^{-64} +26 q^{-66} - q^{-68} -12 q^{-70} +14 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_45.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+3 q^9-3 q^7+4 q^5-3 q^3+q+ q^{-1} -3 q^{-3} +4 q^{-5} -3 q^{-7} +3 q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-3 q^{30}-q^{28}+11 q^{26}-9 q^{24}-11 q^{22}+27 q^{20}-10 q^{18}-28 q^{16}+37 q^{14}-37 q^{10}+27 q^8+13 q^6-26 q^4+q^2+19+ q^{-2} -26 q^{-4} +13 q^{-6} +27 q^{-8} -37 q^{-10} +37 q^{-14} -28 q^{-16} -10 q^{-18} +27 q^{-20} -11 q^{-22} -9 q^{-24} +11 q^{-26} - q^{-28} -3 q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+3 q^{61}+q^{59}-7 q^{57}-6 q^{55}+13 q^{53}+21 q^{51}-24 q^{49}-40 q^{47}+27 q^{45}+75 q^{43}-24 q^{41}-118 q^{39}+5 q^{37}+165 q^{35}+29 q^{33}-202 q^{31}-82 q^{29}+222 q^{27}+141 q^{25}-217 q^{23}-191 q^{21}+184 q^{19}+230 q^{17}-134 q^{15}-239 q^{13}+65 q^{11}+228 q^9+6 q^7-193 q^5-75 q^3+140 q+140 q^{-1} -75 q^{-3} -193 q^{-5} +6 q^{-7} +228 q^{-9} +65 q^{-11} -239 q^{-13} -134 q^{-15} +230 q^{-17} +184 q^{-19} -191 q^{-21} -217 q^{-23} +141 q^{-25} +222 q^{-27} -82 q^{-29} -202 q^{-31} +29 q^{-33} +165 q^{-35} +5 q^{-37} -118 q^{-39} -24 q^{-41} +75 q^{-43} +27 q^{-45} -40 q^{-47} -24 q^{-49} +21 q^{-51} +13 q^{-53} -6 q^{-55} -7 q^{-57} + q^{-59} +3 q^{-61} - q^{-63} }[/math]
4	[math]\displaystyle{ q^{104}-3 q^{102}-q^{100}+7 q^{98}+2 q^{96}+2 q^{94}-23 q^{92}-13 q^{90}+33 q^{88}+29 q^{86}+28 q^{84}-89 q^{82}-93 q^{80}+65 q^{78}+135 q^{76}+160 q^{74}-190 q^{72}-334 q^{70}-22 q^{68}+317 q^{66}+553 q^{64}-155 q^{62}-748 q^{60}-450 q^{58}+363 q^{56}+1210 q^{54}+300 q^{52}-1027 q^{50}-1229 q^{48}-76 q^{46}+1737 q^{44}+1165 q^{42}-755 q^{40}-1891 q^{38}-951 q^{36}+1634 q^{34}+1906 q^{32}+44 q^{30}-1906 q^{28}-1709 q^{26}+898 q^{24}+2002 q^{22}+864 q^{20}-1261 q^{18}-1886 q^{16}-49 q^{14}+1461 q^{12}+1334 q^{10}-332 q^8-1524 q^6-873 q^4+613 q^2+1459+613 q^{-2} -873 q^{-4} -1524 q^{-6} -332 q^{-8} +1334 q^{-10} +1461 q^{-12} -49 q^{-14} -1886 q^{-16} -1261 q^{-18} +864 q^{-20} +2002 q^{-22} +898 q^{-24} -1709 q^{-26} -1906 q^{-28} +44 q^{-30} +1906 q^{-32} +1634 q^{-34} -951 q^{-36} -1891 q^{-38} -755 q^{-40} +1165 q^{-42} +1737 q^{-44} -76 q^{-46} -1229 q^{-48} -1027 q^{-50} +300 q^{-52} +1210 q^{-54} +363 q^{-56} -450 q^{-58} -748 q^{-60} -155 q^{-62} +553 q^{-64} +317 q^{-66} -22 q^{-68} -334 q^{-70} -190 q^{-72} +160 q^{-74} +135 q^{-76} +65 q^{-78} -93 q^{-80} -89 q^{-82} +28 q^{-84} +29 q^{-86} +33 q^{-88} -13 q^{-90} -23 q^{-92} +2 q^{-94} +2 q^{-96} +7 q^{-98} - q^{-100} -3 q^{-102} + q^{-104} }[/math]
5	[math]\displaystyle{ -q^{155}+3 q^{153}+q^{151}-7 q^{149}-2 q^{147}+2 q^{145}+8 q^{143}+15 q^{141}+4 q^{139}-33 q^{137}-43 q^{135}-q^{133}+58 q^{131}+95 q^{129}+41 q^{127}-98 q^{125}-222 q^{123}-135 q^{121}+165 q^{119}+406 q^{117}+333 q^{115}-139 q^{113}-712 q^{111}-764 q^{109}+23 q^{107}+1089 q^{105}+1418 q^{103}+432 q^{101}-1404 q^{99}-2455 q^{97}-1334 q^{95}+1492 q^{93}+3692 q^{91}+2874 q^{89}-1004 q^{87}-4953 q^{85}-5068 q^{83}-319 q^{81}+5824 q^{79}+7722 q^{77}+2646 q^{75}-5867 q^{73}-10372 q^{71}-5922 q^{69}+4731 q^{67}+12493 q^{65}+9707 q^{63}-2337 q^{61}-13500 q^{59}-13387 q^{57}-1098 q^{55}+13073 q^{53}+16317 q^{51}+5018 q^{49}-11250 q^{47}-17911 q^{45}-8741 q^{43}+8263 q^{41}+17990 q^{39}+11748 q^{37}-4766 q^{35}-16633 q^{33}-13554 q^{31}+1176 q^{29}+14178 q^{27}+14228 q^{25}+2011 q^{23}-11137 q^{21}-13854 q^{19}-4629 q^{17}+7850 q^{15}+12833 q^{13}+6722 q^{11}-4616 q^9-11487 q^7-8436 q^5+1525 q^3+9988 q+9988 q^{-1} +1525 q^{-3} -8436 q^{-5} -11487 q^{-7} -4616 q^{-9} +6722 q^{-11} +12833 q^{-13} +7850 q^{-15} -4629 q^{-17} -13854 q^{-19} -11137 q^{-21} +2011 q^{-23} +14228 q^{-25} +14178 q^{-27} +1176 q^{-29} -13554 q^{-31} -16633 q^{-33} -4766 q^{-35} +11748 q^{-37} +17990 q^{-39} +8263 q^{-41} -8741 q^{-43} -17911 q^{-45} -11250 q^{-47} +5018 q^{-49} +16317 q^{-51} +13073 q^{-53} -1098 q^{-55} -13387 q^{-57} -13500 q^{-59} -2337 q^{-61} +9707 q^{-63} +12493 q^{-65} +4731 q^{-67} -5922 q^{-69} -10372 q^{-71} -5867 q^{-73} +2646 q^{-75} +7722 q^{-77} +5824 q^{-79} -319 q^{-81} -5068 q^{-83} -4953 q^{-85} -1004 q^{-87} +2874 q^{-89} +3692 q^{-91} +1492 q^{-93} -1334 q^{-95} -2455 q^{-97} -1404 q^{-99} +432 q^{-101} +1418 q^{-103} +1089 q^{-105} +23 q^{-107} -764 q^{-109} -712 q^{-111} -139 q^{-113} +333 q^{-115} +406 q^{-117} +165 q^{-119} -135 q^{-121} -222 q^{-123} -98 q^{-125} +41 q^{-127} +95 q^{-129} +58 q^{-131} - q^{-133} -43 q^{-135} -33 q^{-137} +4 q^{-139} +15 q^{-141} +8 q^{-143} +2 q^{-145} -2 q^{-147} -7 q^{-149} + q^{-151} +3 q^{-153} - q^{-155} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}+q^{14}+2 q^{12}-2 q^{10}+3 q^8-2 q^4+2 q^2-3+2 q^{-2} -2 q^{-4} +3 q^{-8} -2 q^{-10} +2 q^{-12} + q^{-14} - q^{-16} }[/math]
1,1	[math]\displaystyle{ q^{44}-6 q^{42}+20 q^{40}-50 q^{38}+105 q^{36}-198 q^{34}+336 q^{32}-524 q^{30}+755 q^{28}-1002 q^{26}+1248 q^{24}-1436 q^{22}+1518 q^{20}-1454 q^{18}+1210 q^{16}-784 q^{14}+187 q^{12}+522 q^{10}-1274 q^8+1984 q^6-2565 q^4+2950 q^2-3078+2950 q^{-2} -2565 q^{-4} +1984 q^{-6} -1274 q^{-8} +522 q^{-10} +187 q^{-12} -784 q^{-14} +1210 q^{-16} -1454 q^{-18} +1518 q^{-20} -1436 q^{-22} +1248 q^{-24} -1002 q^{-26} +755 q^{-28} -524 q^{-30} +336 q^{-32} -198 q^{-34} +105 q^{-36} -50 q^{-38} +20 q^{-40} -6 q^{-42} + q^{-44} }[/math]
2,0	[math]\displaystyle{ q^{42}-q^{40}-3 q^{38}+6 q^{34}+3 q^{32}-10 q^{30}-2 q^{28}+15 q^{26}+4 q^{24}-19 q^{22}-5 q^{20}+21 q^{18}+4 q^{16}-24 q^{14}+20 q^{10}-6 q^8-12 q^6+8 q^4+6 q^2-6+6 q^{-2} +8 q^{-4} -12 q^{-6} -6 q^{-8} +20 q^{-10} -24 q^{-14} +4 q^{-16} +21 q^{-18} -5 q^{-20} -19 q^{-22} +4 q^{-24} +15 q^{-26} -2 q^{-28} -10 q^{-30} +3 q^{-32} +6 q^{-34} -3 q^{-38} - q^{-40} + q^{-42} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-3 q^{32}+q^{30}+6 q^{28}-12 q^{26}+5 q^{24}+16 q^{22}-23 q^{20}+6 q^{18}+26 q^{16}-30 q^{14}+27 q^{10}-23 q^8-7 q^6+18 q^4-q^2-8- q^{-2} +18 q^{-4} -7 q^{-6} -23 q^{-8} +27 q^{-10} -30 q^{-14} +26 q^{-16} +6 q^{-18} -23 q^{-20} +16 q^{-22} +5 q^{-24} -12 q^{-26} +6 q^{-28} + q^{-30} -3 q^{-32} + q^{-34} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}+q^{19}+2 q^{15}-2 q^{13}+4 q^{11}-q^9+2 q^7-2 q^5+q^3-2 q-2 q^{-1} + q^{-3} -2 q^{-5} +2 q^{-7} - q^{-9} +4 q^{-11} -2 q^{-13} +2 q^{-15} + q^{-19} - q^{-21} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+3 q^{32}-7 q^{30}+12 q^{28}-18 q^{26}+25 q^{24}-30 q^{22}+35 q^{20}-34 q^{18}+32 q^{16}-22 q^{14}+10 q^{12}+7 q^{10}-25 q^8+41 q^6-56 q^4+65 q^2-70+65 q^{-2} -56 q^{-4} +41 q^{-6} -25 q^{-8} +7 q^{-10} +10 q^{-12} -22 q^{-14} +32 q^{-16} -34 q^{-18} +35 q^{-20} -30 q^{-22} +25 q^{-24} -18 q^{-26} +12 q^{-28} -7 q^{-30} +3 q^{-32} - q^{-34} }[/math]
1,0	[math]\displaystyle{ q^{56}-3 q^{52}-3 q^{50}+4 q^{48}+9 q^{46}-q^{44}-15 q^{42}-9 q^{40}+17 q^{38}+23 q^{36}-6 q^{34}-32 q^{32}-11 q^{30}+30 q^{28}+30 q^{26}-16 q^{24}-38 q^{22}-3 q^{20}+34 q^{18}+16 q^{16}-25 q^{14}-23 q^{12}+14 q^{10}+24 q^8-7 q^6-23 q^4+3 q^2+25+3 q^{-2} -23 q^{-4} -7 q^{-6} +24 q^{-8} +14 q^{-10} -23 q^{-12} -25 q^{-14} +16 q^{-16} +34 q^{-18} -3 q^{-20} -38 q^{-22} -16 q^{-24} +30 q^{-26} +30 q^{-28} -11 q^{-30} -32 q^{-32} -6 q^{-34} +23 q^{-36} +17 q^{-38} -9 q^{-40} -15 q^{-42} - q^{-44} +9 q^{-46} +4 q^{-48} -3 q^{-50} -3 q^{-52} + q^{-56} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+14 q^{72}-12 q^{70}-q^{68}+26 q^{66}-51 q^{64}+77 q^{62}-84 q^{60}+57 q^{58}-82 q^{54}+162 q^{52}-205 q^{50}+193 q^{48}-112 q^{46}-20 q^{44}+163 q^{42}-263 q^{40}+285 q^{38}-209 q^{36}+66 q^{34}+90 q^{32}-201 q^{30}+222 q^{28}-143 q^{26}+10 q^{24}+123 q^{22}-188 q^{20}+146 q^{18}-16 q^{16}-156 q^{14}+293 q^{12}-328 q^{10}+240 q^8-49 q^6-182 q^4+363 q^2-433+363 q^{-2} -182 q^{-4} -49 q^{-6} +240 q^{-8} -328 q^{-10} +293 q^{-12} -156 q^{-14} -16 q^{-16} +146 q^{-18} -188 q^{-20} +123 q^{-22} +10 q^{-24} -143 q^{-26} +222 q^{-28} -201 q^{-30} +90 q^{-32} +66 q^{-34} -209 q^{-36} +285 q^{-38} -263 q^{-40} +163 q^{-42} -20 q^{-44} -112 q^{-46} +193 q^{-48} -205 q^{-50} +162 q^{-52} -82 q^{-54} +57 q^{-58} -84 q^{-60} +77 q^{-62} -51 q^{-64} +26 q^{-66} - q^{-68} -12 q^{-70} +14 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }[/math]

.

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["10 45"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+7 t^2-21 t+31-21 t^{-1} +7 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6+z^4-2 z^2+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 89, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^5+4 q^4-7 q^3+11 q^2-14 q+15-14 q^{-1} +11 q^{-2} -7 q^{-3} +4 q^{-4} - q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^6+2 a^2 z^4+2 z^4 a^{-2} -3 z^4-a^4 z^2+3 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} -6 z^2+2 a^2+2 a^{-2} -3 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a z^9+z^9 a^{-1} +4 a^2 z^8+4 z^8 a^{-2} +8 z^8+6 a^3 z^7+14 a z^7+14 z^7 a^{-1} +6 z^7 a^{-3} +4 a^4 z^6+3 a^2 z^6+3 z^6 a^{-2} +4 z^6 a^{-4} -2 z^6+a^5 z^5-10 a^3 z^5-31 a z^5-31 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -7 a^4 z^4-17 a^2 z^4-17 z^4 a^{-2} -7 z^4 a^{-4} -20 z^4-a^5 z^3+5 a^3 z^3+21 a z^3+21 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +3 a^4 z^2+12 a^2 z^2+12 z^2 a^{-2} +3 z^2 a^{-4} +18 z^2-a^3 z-5 a z-5 z a^{-1} -z a^{-3} -2 a^2-2 a^{-2} -3 }[/math]

Vassiliev invariants

V2 and V3:

(-2, 0)

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 [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{116}{3} }[/math] [math]\displaystyle{ \frac{28}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{928}{3} }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ -\frac{4471}{15} }[/math] [math]\displaystyle{ -\frac{116}{15} }[/math] [math]\displaystyle{ -\frac{7084}{45} }[/math] [math]\displaystyle{ \frac{151}{9} }[/math] [math]\displaystyle{ -\frac{631}{15} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 10 45. 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	5	χ
11											1	-1
9										3		3
7									4	1		-3
5								7	3			4
3							7	4				-3
1						8	7					1
-1					7	8						1
-3				4	7							-3
-5			3	7								4
-7		1	4									-3
-9		3										3
-11	1											-1

Computer Talk

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

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 45]]
Out[2]=	10
In[3]:=	PD[Knot[10, 45]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 6, 13, 5], X[10, 3, 11, 4], X[2, 11, 3, 12], X[20, 14, 1, 13], X[14, 7, 15, 8], X[6, 19, 7, 20], X[18, 15, 19, 16], X[16, 10, 17, 9], X[8, 18, 9, 17]]
In[4]:=	GaussCode[Knot[10, 45]]
Out[4]=	GaussCode[1, -4, 3, -1, 2, -7, 6, -10, 9, -3, 4, -2, 5, -6, 8, -9, 10, -8, 7, -5]
In[5]:=	BR[Knot[10, 45]]
Out[5]=	BR[5, {-1, 2, -1, 2, -3, 2, -3, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[10, 45]][t]
Out[6]=	-3 7 21 2 3 31 - t + -- - -- - 21 t + 7 t - t 2 t t
In[7]:=	Conway[Knot[10, 45]][z]
Out[7]=	2 4 6 1 - 2 z + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 45]}
In[9]:=	{KnotDet[Knot[10, 45]], KnotSignature[Knot[10, 45]]}
Out[9]=	{89, 0}
In[10]:=	J=Jones[Knot[10, 45]][q]
Out[10]=	-5 4 7 11 14 2 3 4 5 15 - q + -- - -- + -- - -- - 14 q + 11 q - 7 q + 4 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 45]}
In[12]:=	A2Invariant[Knot[10, 45]][q]
Out[12]=	-16 -14 2 2 3 2 2 2 4 8 -3 - q + q + --- - --- + -- - -- + -- + 2 q - 2 q + 3 q - 12 10 8 4 2 q q q q q 10 12 14 16 2 q + 2 q + q - q
In[13]:=	Kauffman[Knot[10, 45]][a, z]
Out[13]=	2 2 2 2 z 5 z 3 2 3 z 12 z -3 - -- - 2 a - -- - --- - 5 a z - a z + 18 z + ---- + ----- + 2 3 a 4 2 a a a a 3 3 3 2 2 4 2 z 5 z 21 z 3 3 3 5 3 12 a z + 3 a z - -- + ---- + ----- + 21 a z + 5 a z - a z - 5 3 a a a 4 4 5 5 5 4 7 z 17 z 2 4 4 4 z 10 z 31 z 20 z - ---- - ----- - 17 a z - 7 a z + -- - ----- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 4 z 3 z 2 6 4 6 31 a z - 10 a z + a z - 2 z + ---- + ---- + 3 a z + 4 a z + 4 2 a a 7 7 8 9 6 z 14 z 7 3 7 8 4 z 2 8 z 9 ---- + ----- + 14 a z + 6 a z + 8 z + ---- + 4 a z + -- + a z 3 a 2 a a a
In[14]:=	{Vassiliev[2][Knot[10, 45]], Vassiliev[3][Knot[10, 45]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 45]][q, t]
Out[15]=	8 1 3 1 4 3 7 4 - + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 7 7 3 3 2 5 2 5 3 7 3 ---- + --- + 7 q t + 7 q t + 4 q t + 7 q t + 3 q t + 4 q t + 3 q t q t 7 4 9 4 11 5 q t + 3 q t + q t