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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X16,7,17,8 X20,10,21,9 X18,12,19,11 X22,20,11,19 X8,15,9,16 X10,22,5,21 X14,18,15,17 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

q-3 - 2q-2 + 7q-1 - 11 + 17q - 19q2 + 21q3 - 18q4 + 14q5 - 9q6 + 4q7 - q8

A2 (sl(3)) Invariant:

q-10 + q-8 + q-6 + 6q-4 + 2q-2 + 3 + 7q2 - q4 + 5q6 + q10 + 3q12 - 3q14 + 4q16 - 2q18 - 2q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6 - a-6z2 - a-6z4 + 2a-4 + 3a-4z2 + 2a-4z4 + a-4z6 + a-2z-2 + a-2 + a-2z4 + a-2z6 - 2z-2 - 4 - 4z2 - 2z4 + a2z-2 + 2a2 + a2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 5a-8z4 + 4a-8z6 - 2a-7z + 7a-7z3 - 13a-7z5 + 8a-7z7 + 2a-6 - 5a-6z2 + 9a-6z4 - 14a-6z6 + 9a-6z8 - 4a-5z + 17a-5z3 - 21a-5z5 + 4a-5z7 + 5a-5z9 + 4a-4 - 18a-4z2 + 37a-4z4 - 39a-4z6 + 16a-4z8 + a-4z10 + 5a-3z3 - 6a-3z5 - 7a-3z7 + 8a-3z9 + a-2z-2 - 2a-2 - 7a-2z2 + 21a-2z4 - 25a-2z6 + 10a-2z8 + a-2z10 - 2a-1z-1 + 6a-1z - 4a-1z3 - 3a-1z5 - a-1z7 + 3a-1z9 + 2z-2 - 7 + 11z2 - 6z4 - 3z6 + 3z8 - 2az-1 + 4az - 4az5 + 2az7 + a2z-2 - 4a2 + 6a2z2 - 4a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     3   j = 13                   6 1   j = 11                 8 3     j = 9               10 6       j = 7             11 8         j = 5           9 11           j = 3         8 10             j = 1       5 11               j = -1     2 6                 j = -3   1 6                   j = -5   1                     j = -7 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, 425]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 425]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[16, 7, 17, 8], X[20, 10, 21, 9], > X[18, 12, 19, 11], X[22, 20, 11, 19], X[8, 15, 9, 16], X[10, 22, 5, 21], > X[14, 18, 15, 17], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -7, 4, -8}, > {5, -2, 11, -9, 7, -3, 9, -5, 6, -4, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 7 2 3 4 5 6 7 8 -11 + q - -- + - + 17 q - 19 q + 21 q - 18 q + 14 q - 9 q + 4 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 -6 6 2 2 4 6 10 12 14 3 + q + q + q + -- + -- + 7 q - q + 5 q + q + 3 q - 3 q + 4 2 q q 16 18 20 22 24 > 4 q - 2 q - 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 425]][a, z]
Out[8]=	2 2 2 -6 2 -2 2 2 1 a 2 z 3 z 2 2 -4 - a + -- + a + 2 a - -- + ----- + -- - 4 z - -- + ---- + a z - 4 2 2 2 2 6 4 a z a z z a a 4 4 4 6 6 4 z 2 z z z z > 2 z - -- + ---- + -- + -- + -- 6 4 2 4 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 425]][a, z]
Out[9]=	2 2 4 2 2 2 1 a 2 2 a 2 z 4 z 6 z -7 + -- + -- - -- - 4 a + -- + ----- + -- - --- - --- - --- - --- + --- + 6 4 2 2 2 2 2 a z z 7 5 a a a a z a z z a a 2 2 2 2 3 3 3 2 z 5 z 18 z 7 z 2 2 z 7 z 17 z > 4 a z + 11 z + -- - ---- - ----- - ---- + 6 a z - -- + ---- + ----- + 8 6 4 2 9 7 5 a a a a a a a 3 3 4 4 4 4 5 5 5 z 4 z 4 5 z 9 z 37 z 21 z 2 4 z 13 z > ---- - ---- - 6 z - ---- + ---- + ----- + ----- - 4 a z + -- - ----- - 3 a 8 6 4 2 9 7 a a a a a a a 5 5 5 6 6 6 6 21 z 6 z 3 z 5 6 4 z 14 z 39 z 25 z > ----- - ---- - ---- - 4 a z - 3 z + ---- - ----- - ----- - ----- + 5 3 a 8 6 4 2 a a a a a a 7 7 7 7 8 8 8 2 6 8 z 4 z 7 z z 7 8 9 z 16 z 10 z > a z + ---- + ---- - ---- - -- + 2 a z + 3 z + ---- + ----- + ----- + 7 5 3 a 6 4 2 a a a a a a 9 9 9 10 10 5 z 8 z 3 z z z > ---- + ---- + ---- + --- + --- 5 3 a 4 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 6 2 6 5 q 3 11 q + 8 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 10 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 9 q t + 11 q t + 11 q t + 8 q t + 10 q t + 6 q t + 8 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a425

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