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DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X16,5,17,6 X12,15,5,16 X22,20,13,19 X18,8,19,7 X10,14,11,13 X8,21,9,22 X20,9,21,10 X2,11,3,12 X4,18,1,17

Gauss Code:

{{1, -10, 2, -11}, {3, -1, 6, -8, 9, -7, 10, -4}, {7, -2, 4, -3, 11, -6, 5, -9, 8, -5}}

Jones Polynomial:

q-6 - 4q-5 + 9q-4 - 13q-3 + 19q-2 - 20q-1 + 22 - 18q + 13q2 - 8q3 + 4q4 - q5

A2 (sl(3)) Invariant:

q-18 - q-16 + 3q-12 + 8q-8 + 4q-6 + 4q-4 + 6q-2 - 2 + 5q2 - 3q4 + q6 + 2q8 - 2q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- 2a-2z2 - 3a-2z4 - a-2z6 + z-2 + 2 + 6z2 + 9z4 + 5z6 + z8 - 2a2z-2 - 3a2 - 7a2z2 - 7a2z4 - 2a2z6 + a4z-2 + a4 + 2a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + a-4z2 - 6a-4z4 + 4a-4z6 + 4a-3z3 - 11a-3z5 + 7a-3z7 - 3a-2z2 + 7a-2z4 - 12a-2z6 + 8a-2z8 + 6a-1z3 - 5a-1z5 - 4a-1z7 + 6a-1z9 - z-2 + 3 - 14z2 + 39z4 - 37z6 + 12z8 + 2z10 + 2az-1 - 3az + az3 + 14az5 - 24az7 + 12az9 - 2a2z-2 + 5a2 - 18a2z2 + 40a2z4 - 39a2z6 + 11a2z8 + 2a2z10 + 2a3z-1 - 3a3z + 4a3z3 - 2a3z5 - 9a3z7 + 6a3z9 - a4z-2 + 3a4 - 7a4z2 + 12a4z4 - 17a4z6 + 7a4z8 + 4a5z3 - 9a5z5 + 4a5z7 + a6z2 - 2a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     3   j = 7                   5 1   j = 5                 8 3     j = 3               10 5       j = 1             12 8         j = -1           11 13           j = -3         8 9             j = -5       5 11               j = -7     4 8                 j = -9   1 6                   j = -11   3                     j = -13 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 459]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 459]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[16, 5, 17, 6], X[12, 15, 5, 16], > X[22, 20, 13, 19], X[18, 8, 19, 7], X[10, 14, 11, 13], X[8, 21, 9, 22], > X[20, 9, 21, 10], X[2, 11, 3, 12], X[4, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {3, -1, 6, -8, 9, -7, 10, -4}, > {7, -2, 4, -3, 11, -6, 5, -9, 8, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-6 4 9 13 19 20 2 3 4 5 22 + q - -- + -- - -- + -- - -- - 18 q + 13 q - 8 q + 4 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 3 8 4 4 6 2 4 6 8 10 -2 + q - q + --- + -- + -- + -- + -- + 5 q - 3 q + q + 2 q - 2 q + 12 8 6 4 2 q q q q q 12 14 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 459]][a, z]
Out[8]=	2 4 2 2 4 -2 2 a a 2 2 z 2 2 4 2 4 2 - 3 a + a + z - ---- + -- + 6 z - ---- - 7 a z + 2 a z + 9 z - 2 2 2 z z a 4 6 3 z 2 4 4 4 6 z 2 6 8 > ---- - 7 a z + a z + 5 z - -- - 2 a z + z 2 2 a a
In[9]:=	Kauffman[Link[11, Alternating, 459]][a, z]
Out[9]=	2 4 3 2 2 4 -2 2 a a 2 a 2 a 3 2 z 3 + 5 a + 3 a - z - ---- - -- + --- + ---- - 3 a z - 3 a z - 14 z + -- - 2 2 z z 4 z z a 2 3 3 3 3 z 2 2 4 2 6 2 z 4 z 6 z 3 3 3 > ---- - 18 a z - 7 a z + a z - -- + ---- + ---- + a z + 4 a z + 2 5 3 a a a a 4 4 5 5 3 4 6 z 7 z 2 4 4 4 6 4 z > 4 a z + 39 z - ---- + ---- + 40 a z + 12 a z - 2 a z + -- - 4 2 5 a a a 5 5 6 6 11 z 5 z 5 3 5 5 5 6 4 z 12 z > ----- - ---- + 14 a z - 2 a z - 9 a z - 37 z + ---- - ----- - 3 a 4 2 a a a 7 7 2 6 4 6 6 6 7 z 4 z 7 3 7 5 7 > 39 a z - 17 a z + a z + ---- - ---- - 24 a z - 9 a z + 4 a z + 3 a a 8 9 8 8 z 2 8 4 8 6 z 9 3 9 10 > 12 z + ---- + 11 a z + 7 a z + ---- + 12 a z + 6 a z + 2 z + 2 a a 2 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 3 1 6 4 8 5 11 -- + 12 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 8 9 11 3 3 2 5 2 5 3 > ----- + ---- + --- + 8 q t + 10 q t + 5 q t + 8 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a459

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