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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 10q-5/2 - 17q-3/2 + 22q-1/2 - 27q1/2 + 26q3/2 - 23q5/2 + 17q7/2 - 10q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-7z3 - a-7z5 - a-6z2 + 4a-6z4 - 4a-6z6 + 3a-5z - 9a-5z3 + 13a-5z5 - 9a-5z7 + 4a-4z2 - 12a-4z4 + 18a-4z6 - 12a-4z8 + a-3z-1 + a-3z - 13a-3z3 + 15a-3z5 + 3a-3z7 - 9a-3z9 - a-2 + 9a-2z2 - 33a-2z4 + 46a-2z6 - 18a-2z8 - 3a-2z10 + a-1z-1 - 5a-1z + 2a-1z3 - 7a-1z5 + 27a-1z7 - 17a-1z9 + 9z2 - 31z4 + 42z6 - 14z8 - 3z10 - 2az + 11az7 - 8az9 + 4a2z2 - 12a2z4 + 17a2z6 - 8a2z8 + a3z - 5a3z3 + 8a3z5 - 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                   7 1   j = 8                 10 3     j = 6               13 7       j = 4             14 11         j = 2           13 12           j = 0         10 15             j = -2       7 12               j = -4     3 10                 j = -6   1 7                   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, 381]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 381]]
Out[4]=	PD[X[12, 1, 13, 2], X[20, 13, 21, 14], X[10, 21, 1, 22], X[14, 4, 15, 3], > X[18, 6, 19, 5], X[6, 11, 7, 12], X[22, 7, 11, 8], X[4, 16, 5, 15], > X[8, 18, 9, 17], X[16, 10, 17, 9], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 4, -8, 5, -6, 7, -9, 10, -3}, > {6, -1, 2, -4, 8, -10, 9, -5, 11, -2, 3, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 10 17 22 3/2 5/2 q - ---- + ---- - ---- + ------- - 27 Sqrt[q] + 26 q - 23 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 17 q - 10 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -10 4 4 2 2 4 6 8 10 6 - q + q + q - -- + -- - -- - 3 q + 6 q - 3 q + 2 q + 3 q - 8 6 4 q q q 12 14 16 18 20 > 4 q + 4 q - q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 381]][a, z]
Out[8]=	3 3 5 5 1 1 z 3 z 3 z 3 z 3 3 3 2 z 2 z -(----) + --- - -- + --- - a z - -- + ---- + 3 a z - a z + ---- - ---- + 3 a z 5 a 5 3 3 a a z a a a a 7 5 z > 2 a z - -- a
In[9]:=	Kauffman[Link[11, Alternating, 381]][a, z]
Out[9]=	2 2 2 -2 1 1 3 z z 5 z 3 2 z 4 z 9 z -a + ---- + --- + --- + -- - --- - 2 a z + a z + 9 z - -- + ---- + ---- + 3 a z 5 3 a 6 4 2 a z a a a a a 3 3 3 3 4 2 2 4 2 z 9 z 13 z 2 z 3 3 4 4 z > 4 a z - a z + -- - ---- - ----- + ---- - 5 a z - 31 z + ---- - 7 5 3 a 6 a a a a 4 4 5 5 5 5 12 z 33 z 2 4 4 4 z 13 z 15 z 7 z 3 5 > ----- - ----- - 12 a z + 2 a z - -- + ----- + ----- - ---- + 8 a z + 4 2 7 5 3 a a a a a a 6 6 6 7 7 7 6 4 z 18 z 46 z 2 6 4 6 9 z 3 z 27 z > 42 z - ---- + ----- + ----- + 17 a z - a z - ---- + ---- + ----- + 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 12 z 18 z 2 8 9 z 17 z > 11 a z - 4 a z - 14 z - ----- - ----- - 8 a z - ---- - ----- - 4 2 3 a a a a 10 9 10 3 z > 8 a z - 3 z - ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 7 3 10 7 10 15 + 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 12 2 4 4 2 6 2 6 3 8 3 > ---- + 12 q t + 14 q t + 11 q t + 13 q t + 7 q t + 10 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a381

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