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DrawMorseLink

PD Presentation:

X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X18,12,19,11 X20,9,21,10 X10,19,5,20 X4,15,1,16 X22,18,11,17 X16,22,17,21

Gauss Code:

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

Jones Polynomial:

- q-4 + 4q-3 - 7q-2 + 14q-1 - 16 + 21q - 20q2 + 18q3 - 14q4 + 8q5 - 4q6 + q7

A2 (sl(3)) Invariant:

- q-12 + q-10 + 2q-8 + 2q-6 + 9q-4 + 5q-2 + 9 + 7q2 + 4q6 - 6q8 - 3q12 - 3q14 + 2q16 - 2q18 + q20

HOMFLY-PT Polynomial:

- a-4z-2 - a-4 + 2a-4z2 + 3a-4z4 + a-4z6 + 4a-2z-2 + 4a-2 - 5a-2z2 - 9a-2z4 - 5a-2z6 - a-2z8 - 5z-2 - 4 + 5z2 + 7z4 + 2z6 + 2a2z-2 + a2 - 2a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - 2a-7z3 + 4a-7z5 - 6a-6z4 + 8a-6z6 - a-5z-1 + a-5z + 6a-5z3 - 16a-5z5 + 12a-5z7 + a-4z-2 - 8a-4z2 + 18a-4z4 - 24a-4z6 + 13a-4z8 - 5a-3z-1 + 5a-3z + 15a-3z3 - 19a-3z5 - 5a-3z7 + 8a-3z9 + 4a-2z-2 - 2a-2 - 24a-2z2 + 68a-2z4 - 67a-2z6 + 18a-2z8 + 2a-2z10 - 9a-1z-1 + 9a-1z + 6a-1z3 + 11a-1z5 - 32a-1z7 + 13a-1z9 + 5z-2 - 4 - 23z2 + 61z4 - 50z6 + 9z8 + 2z10 - 5az-1 + 5az + az3 + 7az5 - 14az7 + 5az9 + 2a2z-2 - 3a2 - 7a2z2 + 18a2z4 - 15a2z6 + 4a2z8 + 2a3z3 - 3a3z5 + a3z7

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 = 15                       1 j = 13                     3   j = 11                   5 1   j = 9                 9 3     j = 7               9 5       j = 5             11 9         j = 3           10 9           j = 1         8 13             j = -1       6 8               j = -3     3 10                 j = -5   1 4                   j = -7   3                     j = -9 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, 400]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 400]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 6, 13, 5], X[8, 4, 9, 3], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[18, 12, 19, 11], X[20, 9, 21, 10], X[10, 19, 5, 20], > X[4, 15, 1, 16], X[22, 18, 11, 17], X[16, 22, 17, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -9}, {2, -1, 5, -3, 7, -8}, > {6, -2, 4, -5, 9, -11, 10, -6, 8, -7, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-4 4 7 14 2 3 4 5 6 7 -16 - q + -- - -- + -- + 21 q - 20 q + 18 q - 14 q + 8 q - 4 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 2 2 9 5 2 6 8 12 14 9 - q + q + -- + -- + -- + -- + 7 q + 4 q - 6 q - 3 q - 3 q + 8 6 4 2 q q q q 16 18 20 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 400]][a, z]
Out[8]=	2 2 2 -4 4 2 5 1 4 2 a 2 2 z 5 z 2 2 -4 - a + -- + a - -- - ----- + ----- + ---- + 5 z + ---- - ---- - 2 a z + 2 2 4 2 2 2 2 4 2 a z a z a z z a a 4 4 6 6 8 4 3 z 9 z 2 4 6 z 5 z z > 7 z + ---- - ---- - a z + 2 z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 400]][a, z]
Out[9]=	2 2 2 5 1 4 2 a 1 5 9 5 a z -4 - -- - 3 a + -- + ----- + ----- + ---- - ---- - ---- - --- - --- + -- + 2 2 4 2 2 2 2 5 3 a z z 5 a z a z a z z a z a z a 2 2 3 3 3 5 z 9 z 2 8 z 24 z 2 2 2 z 6 z 15 z > --- + --- + 5 a z - 23 z - ---- - ----- - 7 a z - ---- + ---- + ----- + 3 a 4 2 7 5 3 a a a a a a 3 4 4 4 4 6 z 3 3 3 4 z 6 z 18 z 68 z 2 4 > ---- + a z + 2 a z + 61 z + -- - ---- + ----- + ----- + 18 a z + a 8 6 4 2 a a a a 5 5 5 5 6 6 4 z 16 z 19 z 11 z 5 3 5 6 8 z 24 z > ---- - ----- - ----- + ----- + 7 a z - 3 a z - 50 z + ---- - ----- - 7 5 3 a 6 4 a a a a a 6 7 7 7 8 67 z 2 6 12 z 5 z 32 z 7 3 7 8 13 z > ----- - 15 a z + ----- - ---- - ----- - 14 a z + a z + 9 z + ----- + 2 5 3 a 4 a a a a 8 9 9 10 18 z 2 8 8 z 13 z 9 10 2 z > ----- + 4 a z + ---- + ----- + 5 a z + 2 z + ----- 2 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 1 4 3 10 6 8 13 q + 10 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 8 q 3 5 5 2 7 2 7 3 9 3 9 4 > --- + 9 q t + 11 q t + 9 q t + 9 q t + 5 q t + 9 q t + 3 q t + t 11 4 11 5 13 5 15 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a400

L11a399 L11a401