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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 6q-9/2 + 10q-7/2 - 14q-5/2 + 15q-3/2 - 16q-1/2 + 13q1/2 - 10q3/2 + 6q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 - q-16 + 2q-14 - 2q-12 + q-10 + 2q-8 - q-6 + 5q-4 - 2q-2 + 4 + 2q6 - 2q8 + q10 - q12

HOMFLY-PT Polynomial:

4a-1z + 8a-1z3 + 5a-1z5 + a-1z7 - az-1 - 9az - 20az3 - 18az5 - 7az7 - az9 + a3z-1 + 4a3z + 8a3z3 + 5a3z5 + a3z7

Kauffman Polynomial:

- a-4z2 + 3a-4z4 - a-4z6 + a-3z - 5a-3z3 + 9a-3z5 - 3a-3z7 + 5a-2z2 - 14a-2z4 + 16a-2z6 - 5a-2z8 - 3a-1z + 10a-1z3 - 18a-1z5 + 16a-1z7 - 5a-1z9 + 10z2 - 26z4 + 17z6 - z8 - 2z10 + az-1 - 9az + 30az3 - 47az5 + 31az7 - 9az9 - a2 + 6a2z2 - 12a2z4 + 6a2z6 - 2a2z10 + a3z-1 - 5a3z + 14a3z3 - 14a3z5 + 8a3z7 - 4a3z9 - a4z2 + 3a4z4 + 3a4z6 - 4a4z8 - a5z + a5z3 + 5a5z5 - 4a5z7 - 3a6z2 + 6a6z4 - 3a6z6 - a7z + 2a7z3 - 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                     2   j = 6                   4 1   j = 4                 6 2     j = 2               7 4       j = 0             9 6         j = -2           7 8           j = -4         7 8             j = -6       4 8               j = -8     2 6                 j = -10   1 4                   j = -12   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, 382]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 382]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[16, 7, 17, 8], > X[18, 10, 19, 9], X[20, 6, 21, 5], X[6, 22, 7, 21], X[22, 18, 11, 17], > X[10, 20, 1, 19], X[8, 11, 9, 12], X[4, 15, 5, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 6, -7, 4, -10, 5, -9}, > {10, -1, 2, -3, 11, -4, 8, -5, 9, -6, 7, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 6 10 14 15 16 -q + ----- - ---- + ---- - ---- + ---- - ------- + 13 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 10 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 2 -10 2 -6 5 2 6 8 10 4 + q - q + --- - --- + q + -- - q + -- - -- + 2 q - 2 q + q - 14 12 8 4 2 q q q q q 12 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 382]][a, z]
Out[8]=	3 3 5 a a 4 z 3 8 z 3 3 3 5 z 5 -(-) + -- + --- - 9 a z + 4 a z + ---- - 20 a z + 8 a z + ---- - 18 a z + z z a a a 7 3 5 z 7 3 7 9 > 5 a z + -- - 7 a z + a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 382]][a, z]
Out[9]=	3 2 2 2 a a z 3 z 3 5 7 2 z 5 z -a + - + -- + -- - --- - 9 a z - 5 a z - a z - a z + 10 z - -- + ---- + z z 3 a 4 2 a a a 3 3 2 2 4 2 6 2 5 z 10 z 3 3 3 5 3 > 6 a z - a z - 3 a z - ---- + ----- + 30 a z + 14 a z + a z + 3 a a 4 4 5 7 3 4 3 z 14 z 2 4 4 4 6 4 9 z > 2 a z - 26 z + ---- - ----- - 12 a z + 3 a z + 6 a z + ---- - 4 2 3 a a a 5 6 6 18 z 5 3 5 5 5 7 5 6 z 16 z > ----- - 47 a z - 14 a z + 5 a z - a z + 17 z - -- + ----- + a 4 2 a a 7 7 2 6 4 6 6 6 3 z 16 z 7 3 7 5 7 > 6 a z + 3 a z - 3 a z - ---- + ----- + 31 a z + 8 a z - 4 a z - 3 a a 8 9 8 5 z 4 8 5 z 9 3 9 10 2 10 > z - ---- - 4 a z - ---- - 9 a z - 4 a z - 2 z - 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 2 1 4 2 6 4 8 9 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 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 7 8 7 2 2 2 4 2 4 3 > ----- + ---- + ---- + 6 t + 7 q t + 4 q t + 6 q t + 2 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a382

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