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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X20,14,17,13 X18,8,19,7 X10,18,11,17 X14,9,15,10 X8,15,9,16 X16,20,5,19 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-6 - 3q-5 + 7q-4 - 11q-3 + 15q-2 - 16q-1 + 16 - 12q + 10q2 - 4q3 + q4

A2 (sl(3)) Invariant:

q-20 + q-18 - 2q-16 + q-14 + q-12 - 3q-10 + 4q-8 + q-6 + 3q-4 + 5q-2 + 1 + 7q2 + 4q6 + 4q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z-2 + 3a-2 + a-2z2 + a-2z4 - 2z-2 - 6 - 7z2 - 3z4 - z6 + a2z-2 + 4a2 + 5a2z2 + 3a2z4 - 2a4 - 3a4z2 + a6

Kauffman Polynomial:

a-4z4 + 4a-3z5 + a-2z-2 - 5a-2 + 8a-2z2 - 12a-2z4 + 10a-2z6 - 2a-1z-1 + 6a-1z + a-1z3 - 15a-1z5 + 12a-1z7 + 2z-2 - 7 + 10z2 - 12z4 - 3z6 + 8z8 - 2az-1 + 2az + 17az3 - 38az5 + 16az7 + 2az9 + a2z-2 - 2a2 + 3a2z2 + 4a2z4 - 21a2z6 + 12a2z8 - 6a3z + 23a3z3 - 27a3z5 + 7a3z7 + 2a3z9 + 4a4z2 - 7a4z6 + 4a4z8 - 2a5z + 7a5z3 - 8a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6

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 j = 9                     1 j = 7                   4 1 j = 5                 6     j = 3               6 4     j = 1             10 6       j = -1           8 8         j = -3         7 8           j = -5       4 8             j = -7     3 7               j = -9   1 5                 j = -11   2                   j = -13 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 155]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 155]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 14, 17, 13], X[18, 8, 19, 7], > X[10, 18, 11, 17], X[14, 9, 15, 10], X[8, 15, 9, 16], X[16, 20, 5, 19], > X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {5, -4, 8, -3}, > {9, -1, 4, -7, 6, -5, 10, -2, 3, -6, 7, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-6 3 7 11 15 16 2 3 4 16 + q - -- + -- - -- + -- - -- - 12 q + 10 q - 4 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 2 -14 -12 3 4 -6 3 5 2 6 1 + q + q - --- + q + q - --- + -- + q + -- + -- + 7 q + 4 q + 16 10 8 4 2 q q q q q 8 10 12 > 4 q - 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 155]][a, z]
Out[8]=	2 2 3 2 4 6 2 1 a 2 z 2 2 4 2 -6 + -- + 4 a - 2 a + a - -- + ----- + -- - 7 z + -- + 5 a z - 3 a z - 2 2 2 2 2 2 a z a z z a 4 4 z 2 4 6 > 3 z + -- + 3 a z - z 2 a
In[9]:=	Kauffman[Link[10, Alternating, 155]][a, z]
Out[9]=	2 5 2 6 2 1 a 2 2 a 6 z 3 -7 - -- - 2 a - a + -- + ----- + -- - --- - --- + --- + 2 a z - 6 a z - 2 2 2 2 2 a z z a a z a z z 2 3 5 2 8 z 2 2 4 2 6 2 z 3 > 2 a z + 10 z + ---- + 3 a z + 4 a z + 3 a z + -- + 17 a z + 2 a a 4 4 5 3 3 5 3 4 z 12 z 2 4 6 4 4 z > 23 a z + 7 a z - 12 z + -- - ----- + 4 a z - 3 a z + ---- - 4 2 3 a a a 5 6 15 z 5 3 5 5 5 6 10 z 2 6 4 6 > ----- - 38 a z - 27 a z - 8 a z - 3 z + ----- - 21 a z - 7 a z + a 2 a 7 6 6 12 z 7 3 7 5 7 8 2 8 4 8 > a z + ----- + 16 a z + 7 a z + 3 a z + 8 z + 12 a z + 4 a z + a 9 3 9 > 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 2 1 5 3 7 4 8 - + 10 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 7 8 8 3 3 2 5 2 7 3 7 4 > ----- + ---- + --- + 6 q t + 6 q t + 4 q t + 6 q t + 4 q t + q t + 3 2 3 q t q t q t 9 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a155

L10a154 L10a156