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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,10,19,9 X22,18,9,17 X8,21,1,22 X20,13,21,14 X14,6,15,5 X16,8,17,7 X6,16,7,15 X4,20,5,19

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 6q-5/2 - 10q-3/2 + 12q-1/2 - 16q1/2 + 15q3/2 - 13q5/2 + 10q7/2 - 6q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-12 + q-10 - 2q-8 + 2q-6 + q-4 + 2q-2 + 6 - q2 + 4q4 - 3q6 - 2q12 + 2q14 - q16 + q18

HOMFLY-PT Polynomial:

- 4a-3z - 8a-3z3 - 5a-3z5 - a-3z7 - a-1z-1 + 8a-1z + 20a-1z3 + 18a-1z5 + 7a-1z7 + a-1z9 + az-1 - 4az - 8az3 - 5az5 - az7

Kauffman Polynomial:

2a-7z3 - a-7z5 - a-6z2 + 6a-6z4 - 3a-6z6 + a-5z - 6a-5z3 + 11a-5z5 - 5a-5z7 + 7a-4z2 - 18a-4z4 + 16a-4z6 - 6a-4z8 - 4a-3z + 11a-3z3 - 20a-3z5 + 14a-3z7 - 5a-3z9 + 13a-2z2 - 29a-2z4 + 15a-2z6 - a-2z8 - 2a-2z10 - a-1z-1 - 10a-1z + 38a-1z3 - 54a-1z5 + 33a-1z7 - 9a-1z9 + 1 + 7z2 - 12z4 + 7z6 + z8 - 2z10 - az-1 - 4az + 13az3 - 13az5 + 11az7 - 4az9 - 4a2z4 + 10a2z6 - 4a2z8 + a3z - 6a3z3 + 9a3z5 - 3a3z7 - 2a4z2 + 3a4z4 - 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                     2   j = 10                   4 1   j = 8                 6 2     j = 6               7 4       j = 4             8 6         j = 2           8 7           j = 0         6 10             j = -2       4 6               j = -4     2 6                 j = -6   1 4                   j = -8   2                     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, 255]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 255]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[18, 10, 19, 9], > X[22, 18, 9, 17], X[8, 21, 1, 22], X[20, 13, 21, 14], X[14, 6, 15, 5], > X[16, 8, 17, 7], X[6, 16, 7, 15], X[4, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 8, -10, 9, -6}, > {4, -1, 2, -3, 7, -8, 10, -9, 5, -4, 11, -7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 6 10 12 3/2 5/2 q - ---- + ---- - ---- + ------- - 16 Sqrt[q] + 15 q - 13 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 10 q - 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 2 2 -4 2 2 4 6 12 14 16 6 - q + q - -- + -- + q + -- - q + 4 q - 3 q - 2 q + 2 q - q + 8 6 2 q q q 18 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 255]][a, z]
Out[8]=	3 3 5 5 1 a 4 z 8 z 8 z 20 z 3 5 z 18 z -(---) + - - --- + --- - 4 a z - ---- + ----- - 8 a z - ---- + ----- - a z z 3 a 3 a 3 a a a a 7 7 9 5 z 7 z 7 z > 5 a z - -- + ---- - a z + -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 255]][a, z]
Out[9]=	2 2 2 1 a z 4 z 10 z 3 2 z 7 z 13 z 1 - --- - - + -- - --- - ---- - 4 a z + a z + 7 z - -- + ---- + ----- - a z z 5 3 a 6 4 2 a a a a a 3 3 3 3 4 4 2 2 z 6 z 11 z 38 z 3 3 3 4 6 z > 2 a z + ---- - ---- + ----- + ----- + 13 a z - 6 a z - 12 z + ---- - 7 5 3 a 6 a a a a 4 4 5 5 5 5 18 z 29 z 2 4 4 4 z 11 z 20 z 54 z 5 > ----- - ----- - 4 a z + 3 a z - -- + ----- - ----- - ----- - 13 a z + 4 2 7 5 3 a a a a a a 6 6 6 7 7 3 5 6 3 z 16 z 15 z 2 6 4 6 5 z 14 z > 9 a z + 7 z - ---- + ----- + ----- + 10 a z - a z - ---- + ----- + 6 4 2 5 3 a a a a a 7 8 8 9 9 33 z 7 3 7 8 6 z z 2 8 5 z 9 z > ----- + 11 a z - 3 a z + z - ---- - -- - 4 a z - ---- - ---- - a 4 2 3 a a a a 10 9 10 2 z > 4 a z - 2 z - ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 4 2 6 4 6 6 10 + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 8 4 > 7 q t + 8 q t + 6 q t + 7 q t + 4 q t + 6 q t + 2 q t + 10 4 10 5 12 5 14 6 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a255

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