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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X22,18,7,17 X18,12,19,11 X20,14,21,13 X12,20,13,19 X14,22,15,21 X6718 X4,15,5,16

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 4q-9/2 + 6q-7/2 - 9q-5/2 + 9q-3/2 - 10q-1/2 + 8q1/2 - 6q3/2 + 4q5/2 - 2q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + 2q-14 + 2q-10 + 3q-8 + q-6 + 4q-4 - q-2 + 2 - q2 - q4 - 2q8 - q12

HOMFLY-PT Polynomial:

2a-1z-1 + 9a-1z + 12a-1z3 + 6a-1z5 + a-1z7 - 5az-1 - 21az - 33az3 - 24az5 - 8az7 - az9 + 3a3z-1 + 9a3z + 12a3z3 + 6a3z5 + a3z7

Kauffman Polynomial:

a-4 - 4a-4z2 + 4a-4z4 - a-4z6 - 5a-3z3 + 7a-3z5 - 2a-3z7 - 2a-2z2 + 5a-2z6 - 2a-2z8 + 2a-1z-1 - 9a-1z + 16a-1z3 - 12a-1z5 + 7a-1z7 - 2a-1z9 - 5 + 20z2 - 19z4 + 6z6 + z8 - z10 + 5az-1 - 25az + 54az3 - 55az5 + 25az7 - 5az9 - 5a2 + 20a2z2 - 29a2z4 + 11a2z6 - a2z10 + 3a3z-1 - 12a3z + 22a3z3 - 26a3z5 + 13a3z7 - 3a3z9 + a4z2 - 9a4z4 + 9a4z6 - 3a4z8 + 3a5z - 8a5z3 + 9a5z5 - 3a5z7 - a6z2 + 5a6z4 - 2a6z6 - a7z + 3a7z3 - a7z5

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 = 10                       1 j = 8                     1   j = 6                   3 1   j = 4                 3 1     j = 2               5 3       j = 0             5 3         j = -2           5 6           j = -4         4 4             j = -6       2 5               j = -8     2 4                 j = -10     2                   j = -12 1 2                     j = -14 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, 200]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 200]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[16, 5, 17, 6], > X[22, 18, 7, 17], X[18, 12, 19, 11], X[20, 14, 21, 13], X[12, 20, 13, 19], > X[14, 22, 15, 21], X[6, 7, 1, 8], X[4, 15, 5, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10}, > {10, -1, 2, -3, 6, -8, 7, -9, 11, -4, 5, -6, 8, -7, 9, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 4 6 9 9 10 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 8 Sqrt[q] - 6 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 9/2 > 4 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 2 3 -6 4 -2 2 4 8 12 2 + q + --- + --- + -- + q + -- - q - q - q - 2 q - q 14 10 8 4 q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 200]][a, z]
Out[8]=	3 3 5 2 5 a 3 a 9 z 3 12 z 3 3 3 6 z --- - --- + ---- + --- - 21 a z + 9 a z + ----- - 33 a z + 12 a z + ---- - a z z z a a a 7 5 3 5 z 7 3 7 9 > 24 a z + 6 a z + -- - 8 a z + a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 200]][a, z]
Out[9]=	3 -4 2 2 5 a 3 a 9 z 3 5 7 -5 + a - 5 a + --- + --- + ---- - --- - 25 a z - 12 a z + 3 a z - a z + a z z z a 2 2 3 3 2 4 z 2 z 2 2 4 2 6 2 5 z 16 z 3 > 20 z - ---- - ---- + 20 a z + a z - a z - ---- + ----- + 54 a z + 4 2 3 a a a a 4 3 3 5 3 7 3 4 4 z 2 4 4 4 > 22 a z - 8 a z + 3 a z - 19 z + ---- - 29 a z - 9 a z + 4 a 5 5 6 6 4 7 z 12 z 5 3 5 5 5 7 5 6 z > 5 a z + ---- - ----- - 55 a z - 26 a z + 9 a z - a z + 6 z - -- + 3 a 4 a a 6 7 7 5 z 2 6 4 6 6 6 2 z 7 z 7 3 7 > ---- + 11 a z + 9 a z - 2 a z - ---- + ---- + 25 a z + 13 a z - 2 3 a a a 8 9 5 7 8 2 z 4 8 2 z 9 3 9 10 2 10 > 3 a z + z - ---- - 3 a z - ---- - 5 a z - 3 a z - z - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 1 2 2 2 4 2 5 5 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 4 4 5 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 3 t + 5 q t + 3 q t + 3 q t + q t + 3 q t + 4 2 4 2 q t q t q t 6 4 8 4 10 5 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a200

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