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DrawMorseLink

PD Presentation:

X6172 X20,7,21,8 X4,21,1,22 X12,6,13,5 X8493 X16,10,17,9 X18,13,19,14 X14,17,15,18 X10,16,11,15 X22,12,5,11 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 9q-5/2 - 16q-3/2 + 21q-1/2 - 25q1/2 + 24q3/2 - 22q5/2 + 16q7/2 - 9q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + q-10 - 3q-8 + 5q-6 - q-4 + 5 - 3q2 + 6q4 - 2q6 + 2q8 + 2q10 - 5q12 + 3q14 - q16 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z - a-5z3 + 3a-3z + 4a-3z3 + 2a-3z5 - a-1z-1 - 4a-1z - 5a-1z3 - 3a-1z5 - a-1z7 + az-1 + 3az + 4az3 + 2az5 - a3z - a3z3

Kauffman Polynomial:

a-7z3 - a-7z5 - 2a-6z2 + 5a-6z4 - 4a-6z6 + 2a-5z - 7a-5z3 + 11a-5z5 - 8a-5z7 - a-4z2 - 4a-4z4 + 12a-4z6 - 10a-4z8 + 6a-3z - 20a-3z3 + 23a-3z5 - 3a-3z7 - 7a-3z9 + 5a-2z2 - 27a-2z4 + 43a-2z6 - 19a-2z8 - 2a-2z10 - a-1z-1 + 8a-1z - 23a-1z3 + 20a-1z5 + 11a-1z7 - 13a-1z9 + 1 + 6z2 - 28z4 + 42z6 - 16z8 - 2z10 - az-1 + 6az - 18az3 + 18az5 + 2az7 - 6az9 + a2z2 - 8a2z4 + 14a2z6 - 7a2z8 + 2a3z - 7a3z3 + 9a3z5 - 4a3z7 - a4z2 + 2a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                       1 j = 12                     3   j = 10                   6 1   j = 8                 10 3     j = 6               12 6       j = 4             12 10         j = 2           13 12           j = 0         10 14             j = -2       6 11               j = -4     3 10                 j = -6   1 6                   j = -8   3                     j = -10 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 13]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 13]]
Out[4]=	PD[X[6, 1, 7, 2], X[20, 7, 21, 8], X[4, 21, 1, 22], X[12, 6, 13, 5], > X[8, 4, 9, 3], X[16, 10, 17, 9], X[18, 13, 19, 14], X[14, 17, 15, 18], > X[10, 16, 11, 15], X[22, 12, 5, 11], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {4, -1, 2, -5, 6, -9, 10, -4, 7, -8, 9, -6, 8, -7, > 11, -2, 3, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 9 16 21 3/2 5/2 q - ---- + ---- - ---- + ------- - 25 Sqrt[q] + 24 q - 22 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 16 q - 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -10 3 5 -4 2 4 6 8 10 5 - q + q + q - -- + -- - q - 3 q + 6 q - 2 q + 2 q + 2 q - 8 6 q q 12 14 16 18 20 > 5 q + 3 q - q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 13]][a, z]
Out[8]=	3 3 3 1 a z 3 z 4 z 3 z 4 z 5 z 3 -(---) + - - -- + --- - --- + 3 a z - a z - -- + ---- - ---- + 4 a z - a z z 5 3 a 5 3 a a a a a 5 5 7 3 3 2 z 3 z 5 z > a z + ---- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 13]][a, z]
Out[9]=	2 2 2 1 a 2 z 6 z 8 z 3 2 2 z z 5 z 1 - --- - - + --- + --- + --- + 6 a z + 2 a z + 6 z - ---- - -- + ---- + a z z 5 3 a 6 4 2 a a a a a 3 3 3 3 2 2 4 2 z 7 z 20 z 23 z 3 3 3 4 > a z - a z + -- - ---- - ----- - ----- - 18 a z - 7 a z - 28 z + 7 5 3 a a a a 4 4 4 5 5 5 5 5 z 4 z 27 z 2 4 4 4 z 11 z 23 z 20 z > ---- - ---- - ----- - 8 a z + 2 a z - -- + ----- + ----- + ----- + 6 4 2 7 5 3 a a a a a a a 6 6 6 5 3 5 6 4 z 12 z 43 z 2 6 4 6 > 18 a z + 9 a z + 42 z - ---- + ----- + ----- + 14 a z - a z - 6 4 2 a a a 7 7 7 8 8 8 z 3 z 11 z 7 3 7 8 10 z 19 z 2 8 > ---- - ---- + ----- + 2 a z - 4 a z - 16 z - ----- - ----- - 7 a z - 5 3 a 4 2 a a a a 9 9 10 7 z 13 z 9 10 2 z > ---- - ----- - 6 a z - 2 z - ----- 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 6 3 10 6 10 14 + 13 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + -- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 11 2 4 4 2 6 2 6 3 8 3 > ---- + 12 q t + 12 q t + 10 q t + 12 q t + 6 q t + 10 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 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 L11a13

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