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DrawMorseLink

PD Presentation:

X8192 X14,9,15,10 X4758 X16,6,17,5 X18,16,19,15 X6,18,1,17 X22,19,7,20 X20,12,21,11 X10,22,11,21 X2,14,3,13 X12,4,13,3

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 4q-5/2 - 10q-3/2 + 16q-1/2 - 22q1/2 + 25q3/2 - 25q5/2 + 21q7/2 - 16q9/2 + 9q11/2 - 4q13/2 + q15/2

A2 (sl(3)) Invariant:

q-10 - 2q-8 + 4q-6 - 3q-2 + 6 - 4q2 + 3q4 - q8 + 4q10 - 4q12 + 4q14 + q16 - 2q18 + 3q20 - q24

HOMFLY-PT Polynomial:

a-7z - a-5z-1 - 3a-5z - 3a-5z3 + 2a-3z-1 + 7a-3z + 7a-3z3 + 3a-3z5 - 2a-1z-1 - 7a-1z - 9a-1z3 - 4a-1z5 - a-1z7 + az-1 + 3az + 2az3 + az5

Kauffman Polynomial:

- a-8z2 + 2a-8z4 - a-8z6 + 2a-7z - 7a-7z3 + 9a-7z5 - 4a-7z7 + a-6z2 - 8a-6z4 + 14a-6z6 - 7a-6z8 - a-5z-1 + 7a-5z - 19a-5z3 + 18a-5z5 + 2a-5z7 - 6a-5z9 + 5a-4z2 - 26a-4z4 + 41a-4z6 - 16a-4z8 - 2a-4z10 - 2a-3z-1 + 13a-3z - 32a-3z3 + 27a-3z5 + 9a-3z7 - 13a-3z9 - a-2 + 6a-2z2 - 28a-2z4 + 45a-2z6 - 20a-2z8 - 2a-2z10 - 2a-1z-1 + 13a-1z - 32a-1z3 + 33a-1z5 - 6a-1z7 - 7a-1z9 + 2z2 - 8z4 + 15z6 - 11z8 - az-1 + 5az - 11az3 + 14az5 - 9az7 - a2z2 + 4a2z4 - 4a2z6 + a3z3 - a3z5

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 = 16                       1 j = 14                     3   j = 12                   6 1   j = 10                 10 3     j = 8               11 6       j = 6             14 10         j = 4           12 12           j = 2         10 13             j = 0       7 13               j = -2     3 9                 j = -4   1 7                   j = -6   3                     j = -8 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, 223]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 223]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 9, 15, 10], X[4, 7, 5, 8], X[16, 6, 17, 5], > X[18, 16, 19, 15], X[6, 18, 1, 17], X[22, 19, 7, 20], X[20, 12, 21, 11], > X[10, 22, 11, 21], X[2, 14, 3, 13], X[12, 4, 13, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 11, -3, 4, -6}, > {3, -1, 2, -9, 8, -11, 10, -2, 5, -4, 6, -5, 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 4 10 16 3/2 5/2 7/2 -q + ---- - ---- + ------- - 22 Sqrt[q] + 25 q - 25 q + 21 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 16 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 4 3 2 4 8 10 12 14 16 6 + q - -- + -- - -- - 4 q + 3 q - q + 4 q - 4 q + 4 q + q - 8 6 2 q q q 18 20 24 > 2 q + 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 223]][a, z]
Out[8]=	3 3 3 1 2 2 a z 3 z 7 z 7 z 3 z 7 z 9 z -(----) + ---- - --- + - + -- - --- + --- - --- + 3 a z - ---- + ---- - ---- + 5 3 a z z 7 5 3 a 5 3 a a z a z a a a a a 5 5 7 3 3 z 4 z 5 z > 2 a z + ---- - ---- + a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 223]][a, z]
Out[9]=	2 -2 1 2 2 a 2 z 7 z 13 z 13 z 2 z -a - ---- - ---- - --- - - + --- + --- + ---- + ---- + 5 a z + 2 z - -- + 5 3 a z z 7 5 3 a 8 a z a z a a a a 2 2 2 3 3 3 3 z 5 z 6 z 2 2 7 z 19 z 32 z 32 z 3 3 3 > -- + ---- + ---- - a z - ---- - ----- - ----- - ----- - 11 a z + a z - 6 4 2 7 5 3 a a a a a a a 4 4 4 4 5 5 5 4 2 z 8 z 26 z 28 z 2 4 9 z 18 z 27 z > 8 z + ---- - ---- - ----- - ----- + 4 a z + ---- + ----- + ----- + 8 6 4 2 7 5 3 a a a a a a a 5 6 6 6 6 33 z 5 3 5 6 z 14 z 41 z 45 z 2 6 > ----- + 14 a z - a z + 15 z - -- + ----- + ----- + ----- - 4 a z - a 8 6 4 2 a a a a 7 7 7 7 8 8 8 9 4 z 2 z 9 z 6 z 7 8 7 z 16 z 20 z 6 z > ---- + ---- + ---- - ---- - 9 a z - 11 z - ---- - ----- - ----- - ---- - 7 5 3 a 6 4 2 5 a a a a a a a 9 9 10 10 13 z 7 z 2 z 2 z > ----- - ---- - ----- - ----- 3 a 4 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 7 3 7 9 2 13 + 10 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 13 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 > 12 q t + 12 q t + 14 q t + 10 q t + 11 q t + 6 q t + 10 q t + 10 5 12 5 12 6 14 6 16 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 L11a223

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