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DrawMorseLink

PD Presentation:

X8192 X10,8,11,7 X12,3,13,4 X18,5,19,6 X16,12,17,11 X14,20,15,19 X20,14,21,13 X2,15,3,16 X22,18,7,17 X4,21,5,22 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 11q-9/2 + 17q-7/2 - 25q-5/2 + 28q-3/2 - 28q-1/2 + 25q1/2 - 18q3/2 + 11q5/2 - 5q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 - q-18 + q-16 + 7q-14 - 2q-12 + 5q-10 + 3q-8 - 4q-6 + 4q-4 - 6q-2 + 3 - 2q2 - 3q4 + 5q6 - 4q8 + q10 + 2q12 - q14

HOMFLY-PT Polynomial:

a-3z3 - 2a-1z - 3a-1z3 - 2a-1z5 + az-1 + 4az + 6az3 + 3az5 + az7 - 3a3z-1 - 6a3z - 4a3z3 - 2a3z5 + 2a5z-1 + a5z + a5z3

Kauffman Polynomial:

a-4z4 - a-4z6 - 4a-3z3 + 9a-3z5 - 5a-3z7 + 2a-2z2 - 13a-2z4 + 21a-2z6 - 10a-2z8 + a-1z - 13a-1z3 + 14a-1z5 + 8a-1z7 - 9a-1z9 + 1 + 14z2 - 49z4 + 62z6 - 21z8 - 3z10 - az-1 - 9az3 + 5az5 + 26az7 - 19az9 + 3a2 + 9a2z2 - 46a2z4 + 62a2z6 - 24a2z8 - 3a2z10 - 3a3z-1 + 7a3z - 16a3z3 + 17a3z5 + 3a3z7 - 10a3z9 + 3a4 - 3a4z2 - 8a4z4 + 18a4z6 - 13a4z8 - 2a5z-1 + 8a5z - 15a5z3 + 16a5z5 - 10a5z7 + 3a6z4 - 4a6z6 + a7z3 - 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                     4   j = 6                   7 1   j = 4                 11 4     j = 2               14 7       j = 0             14 11         j = -2           15 15           j = -4         10 13             j = -6       7 15               j = -8     4 10                 j = -10     7                   j = -12 1 4                     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, 244]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 244]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 8, 11, 7], X[12, 3, 13, 4], X[18, 5, 19, 6], > X[16, 12, 17, 11], X[14, 20, 15, 19], X[20, 14, 21, 13], X[2, 15, 3, 16], > X[22, 18, 7, 17], X[4, 21, 5, 22], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 3, -10, 4, -11}, > {2, -1, 11, -2, 5, -3, 7, -6, 8, -5, 9, -4, 6, -7, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 11 17 25 28 28 -q + ----- - ---- + ---- - ---- + ---- - ------- + 25 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 > 18 q + 11 q - 5 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 7 2 5 3 4 4 6 2 4 3 + q - q + q + --- - --- + --- + -- - -- + -- - -- - 2 q - 3 q + 14 12 10 8 6 4 2 q q q q q q q 6 8 10 12 14 > 5 q - 4 q + q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 244]][a, z]
Out[8]=	3 5 3 3 a 3 a 2 a 2 z 3 5 z 3 z 3 3 3 - - ---- + ---- - --- + 4 a z - 6 a z + a z + -- - ---- + 6 a z - 4 a z + z z z a 3 a a 5 5 3 2 z 5 3 5 7 > a z - ---- + 3 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 244]][a, z]
Out[9]=	3 5 2 2 4 a 3 a 2 a z 3 5 2 2 z 1 + 3 a + 3 a - - - ---- - ---- + - + 7 a z + 8 a z + 14 z + ---- + z z z a 2 a 3 3 2 2 4 2 4 z 13 z 3 3 3 5 3 7 3 > 9 a z - 3 a z - ---- - ----- - 9 a z - 16 a z - 15 a z + a z - 3 a a 4 4 5 5 4 z 13 z 2 4 4 4 6 4 9 z 14 z 5 > 49 z + -- - ----- - 46 a z - 8 a z + 3 a z + ---- + ----- + 5 a z + 4 2 3 a a a a 6 6 3 5 5 5 7 5 6 z 21 z 2 6 4 6 > 17 a z + 16 a z - a z + 62 z - -- + ----- + 62 a z + 18 a z - 4 2 a a 7 7 8 6 6 5 z 8 z 7 3 7 5 7 8 10 z > 4 a z - ---- + ---- + 26 a z + 3 a z - 10 a z - 21 z - ----- - 3 a 2 a a 9 2 8 4 8 9 z 9 3 9 10 2 10 > 24 a z - 13 a z - ---- - 19 a z - 10 a z - 3 z - 3 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	15 1 1 4 7 4 10 7 15 14 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 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 10 13 15 2 2 2 4 2 4 3 > ----- + ---- + ---- + 11 t + 14 q t + 7 q t + 11 q t + 4 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a244

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