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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-1 + 4 - 6q + 10q2 - 13q3 + 16q4 - 15q5 + 14q6 - 10q7 + 7q8 - 3q9 + q10

A2 (sl(3)) Invariant:

- q-2 + 2 + 2q4 + 2q6 - q8 + 4q10 - 2q12 + 4q14 + 3q16 + 3q18 + 6q20 + q22 + 3q24 + q30

HOMFLY-PT Polynomial:

a-8z-2 + 2a-8 + 3a-8z2 + a-8z4 - 2a-6z-2 - 5a-6 - 9a-6z2 - 8a-6z4 - 2a-6z6 + a-4z-2 + a-4 + 5a-4z2 + 8a-4z4 + 5a-4z6 + a-4z8 + 2a-2 - 3a-2z4 - a-2z6

Kauffman Polynomial:

- a-12z2 + a-12z4 - 2a-11z3 + 3a-11z5 - 2a-10 + 6a-10z2 - 8a-10z4 + 6a-10z6 + a-9z + 2a-9z3 - 8a-9z5 + 7a-9z7 - a-8z-2 + 3a-8 - 6a-8z2 + 8a-8z4 - 11a-8z6 + 7a-8z8 + 2a-7z-1 - 8a-7z + 9a-7z3 - 4a-7z5 - 6a-7z7 + 5a-7z9 - 2a-6z-2 + 9a-6 - 24a-6z2 + 37a-6z4 - 29a-6z6 + 5a-6z8 + 2a-6z10 + 2a-5z-1 - 8a-5z - a-5z3 + 29a-5z5 - 33a-5z7 + 10a-5z9 - a-4z-2 + 3a-4 - 13a-4z2 + 36a-4z4 - 28a-4z6 + 2a-4z8 + 2a-4z10 + a-3z - 5a-3z3 + 19a-3z5 - 19a-3z7 + 5a-3z9 - 2a-2 - 2a-2z2 + 16a-2z4 - 16a-2z6 + 4a-2z8 + a-1z3 - 3a-1z5 + a-1z7

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 21                       1 j = 19                     3 1 j = 17                   4     j = 15                 6 3     j = 13               8 4       j = 11             8 7         j = 9           8 7           j = 7         5 8             j = 5       5 8               j = 3     3 7                 j = 1   1 3                   j = -1   3                     j = -3 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, 461]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 461]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[18, 15, 19, 16], X[16, 6, 17, 5], > X[12, 18, 5, 17], X[8, 22, 9, 21], X[20, 8, 21, 7], X[22, 10, 13, 9], > X[10, 14, 11, 13], X[2, 11, 3, 12], X[4, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {4, -1, 7, -6, 8, -9, 10, -5}, > {9, -2, 3, -4, 5, -3, 11, -7, 6, -8}]
In[6]:=	Jones[L][q]
Out[6]=	1 2 3 4 5 6 7 8 9 10 4 - - - 6 q + 10 q - 13 q + 16 q - 15 q + 14 q - 10 q + 7 q - 3 q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 4 6 8 10 12 14 16 18 20 2 - q + 2 q + 2 q - q + 4 q - 2 q + 4 q + 3 q + 3 q + 6 q + 22 24 30 > q + 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 461]][a, z]
Out[8]=	2 2 2 4 4 2 5 -4 2 1 2 1 3 z 9 z 5 z z 8 z -- - -- + a + -- + ----- - ----- + ----- + ---- - ---- + ---- + -- - ---- + 8 6 2 8 2 6 2 4 2 8 6 4 8 6 a a a a z a z a z a a a a a 4 4 6 6 6 8 8 z 3 z 2 z 5 z z z > ---- - ---- - ---- + ---- - -- + -- 4 2 6 4 2 4 a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 461]][a, z]
Out[9]=	-2 3 9 3 2 1 2 1 2 2 z 8 z --- + -- + -- + -- - -- - ----- - ----- - ----- + ---- + ---- + -- - --- - 10 8 6 4 2 8 2 6 2 4 2 7 5 9 7 a a a a a a z a z a z a z a z a a 2 2 2 2 2 2 3 3 3 8 z z z 6 z 6 z 24 z 13 z 2 z 2 z 2 z 9 z > --- + -- - --- + ---- - ---- - ----- - ----- - ---- - ---- + ---- + ---- - 5 3 12 10 8 6 4 2 11 9 7 a a a a a a a a a a a 3 3 3 4 4 4 4 4 4 5 5 z 5 z z z 8 z 8 z 37 z 36 z 16 z 3 z 8 z > -- - ---- + -- + --- - ---- + ---- + ----- + ----- + ----- + ---- - ---- - 5 3 a 12 10 8 6 4 2 11 9 a a a a a a a a a a 5 5 5 5 6 6 6 6 6 7 4 z 29 z 19 z 3 z 6 z 11 z 29 z 28 z 16 z 7 z > ---- + ----- + ----- - ---- + ---- - ----- - ----- - ----- - ----- + ---- - 7 5 3 a 10 8 6 4 2 9 a a a a a a a a a 7 7 7 7 8 8 8 8 9 9 6 z 33 z 19 z z 7 z 5 z 2 z 4 z 5 z 10 z > ---- - ----- - ----- + -- + ---- + ---- + ---- + ---- + ---- + ----- + 7 5 3 a 8 6 4 2 7 5 a a a a a a a a a 9 10 10 5 z 2 z 2 z > ---- + ----- + ----- 3 6 4 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 3 q 3 q 3 q 5 7 7 2 7 q + 5 q + ----- + ---- + -- + --- + ---- + 8 q t + 5 q t + 8 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 15 5 > 8 q t + 7 q t + 8 q t + 7 q t + 8 q t + 4 q t + 6 q t + 15 6 17 6 19 7 19 8 21 8 > 3 q t + 4 q t + 3 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a461

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