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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X22,14,11,13 X10,15,5,16 X20,17,21,18 X16,7,17,8 X8,20,9,19 X18,10,19,9 X14,22,15,21 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-8 + 3q-7 - 6q-6 + 12q-5 - 15q-4 + 19q-3 - 18q-2 + 17q-1 - 12 + 8q - 4q2 + q3

A2 (sl(3)) Invariant:

- q-26 - q-24 + 2q-22 + q-18 + 7q-16 + q-14 + 5q-12 + 5q-10 + q-8 + 5q-6 - 2q-4 + 4q-2 + 1 - 2q2 + 3q4 - 2q6 - q8 + q10

HOMFLY-PT Polynomial:

a-2z2 - 2z2 - 2z4 + a2z-2 + 3a2 + 4a2z2 + 2a2z4 + a2z6 - 2a4z-2 - 5a4 - 5a4z2 - 3a4z4 + a6z-2 + 3a6 + 3a6z2 - a8

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 + 6a-1z3 - 10a-1z5 + 4a-1z7 - z2 + 8z4 - 14z6 + 6z8 + 8az3 - 13az5 - az7 + 4az9 + a2z-2 - 4a2 + a2z2 + 15a2z4 - 27a2z6 + 11a2z8 + a2z10 - 2a3z-1 + 3a3z + 4a3z3 - 8a3z5 - 5a3z7 + 7a3z9 + 2a4z-2 - 9a4 + 16a4z2 - 4a4z4 - 13a4z6 + 9a4z8 + a4z10 - 2a5z-1 + 5a5z + a5z3 - 9a5z5 + 4a5z7 + 3a5z9 + a6z-2 - 8a6 + 18a6z2 - 15a6z4 + 2a6z6 + 4a6z8 + 3a7z - 3a7z3 - 3a7z5 + 4a7z7 - 2a8 + 5a8z2 - 6a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     3   j = 3                   5 1   j = 1                 7 3     j = -1               10 5       j = -3             9 8         j = -5           10 9           j = -7         7 11             j = -9       5 8               j = -11     2 8                 j = -13   1 4                   j = -15   2                     j = -17 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[11, Alternating, 416]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 416]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[22, 14, 11, 13], X[10, 15, 5, 16], > X[20, 17, 21, 18], X[16, 7, 17, 8], X[8, 20, 9, 19], X[18, 10, 19, 9], > X[14, 22, 15, 21], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 6, -7, 8, -4}, > {11, -2, 3, -9, 4, -6, 5, -8, 7, -5, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-8 3 6 12 15 19 18 17 2 3 -12 - q + -- - -- + -- - -- + -- - -- + -- + 8 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 2 -18 7 -14 5 5 -8 5 2 4 1 - q - q + --- + q + --- + q + --- + --- + q + -- - -- + -- - 22 16 12 10 6 4 2 q q q q q q q 2 4 6 8 10 > 2 q + 3 q - 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 416]][a, z]
Out[8]=	2 4 6 2 2 4 6 8 a 2 a a 2 z 2 2 4 2 3 a - 5 a + 3 a - a + -- - ---- + -- - 2 z + -- + 4 a z - 5 a z + 2 2 2 2 z z z a 6 2 4 2 4 4 4 2 6 > 3 a z - 2 z + 2 a z - 3 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 416]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 a 2 a a 2 a 2 a 3 5 -4 a - 9 a - 8 a - 2 a + -- + ---- + -- - ---- - ---- + 3 a z + 5 a z + 2 2 2 z z z z z 2 3 7 9 2 z 2 2 4 2 6 2 8 2 6 z > 3 a z + a z - z + -- + a z + 16 a z + 18 a z + 5 a z + ---- + 2 a a 4 3 3 3 5 3 7 3 9 3 4 2 z 2 4 > 8 a z + 4 a z + a z - 3 a z - 2 a z + 8 z - ---- + 15 a z - 2 a 5 4 4 6 4 8 4 10 z 5 3 5 5 5 > 4 a z - 15 a z - 6 a z - ----- - 13 a z - 8 a z - 9 a z - a 6 7 5 9 5 6 z 2 6 4 6 6 6 8 6 > 3 a z + a z - 14 z + -- - 27 a z - 13 a z + 2 a z + 3 a z + 2 a 7 4 z 7 3 7 5 7 7 7 8 2 8 4 8 > ---- - a z - 5 a z + 4 a z + 4 a z + 6 z + 11 a z + 9 a z + a 6 8 9 3 9 5 9 2 10 4 10 > 4 a z + 4 a z + 7 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 10 1 2 1 4 2 8 5 8 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 7 11 10 9 9 5 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 7 q t + 3 q t + 5 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a416

L11a415 L11a417