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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-17/2 - 4q-15/2 + 9q-13/2 - 14q-11/2 + 18q-9/2 - 21q-7/2 + 19q-5/2 - 17q-3/2 + 12q-1/2 - 7q1/2 + 3q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-26 + q-24 + q-22 - 3q-20 + 3q-18 - 2q-16 + 3q-12 - 2q-10 + 6q-8 - q-6 + 3q-4 + 2q-2 - 3 + 2q2 - q4 + q8

HOMFLY-PT Polynomial:

- 2a-1z - a-1z3 - az-1 + 2az + 5az3 + 2az5 + a3z-1 - a3z - 3a3z3 - 3a3z5 - a3z7 + a5z + 4a5z3 + 2a5z5 - a7z - a7z3

Kauffman Polynomial:

2a-1z - 5a-1z3 + 4a-1z5 - a-1z7 + 5z2 - 13z4 + 11z6 - 3z8 + az-1 - az - 2az3 - 6az5 + 11az7 - 4az9 - a2 + 8a2z2 - 27a2z4 + 26a2z6 - 4a2z8 - 2a2z10 + a3z-1 - 5a3z + 3a3z3 - 11a3z5 + 25a3z7 - 11a3z9 + 6a4z2 - 25a4z4 + 39a4z6 - 13a4z8 - 2a4z10 - 11a5z3 + 21a5z5 - 7a5z9 - 2a6z4 + 15a6z6 - 12a6z8 + 2a7z - 10a7z3 + 18a7z5 - 13a7z7 - 3a8z2 + 8a8z4 - 9a8z6 + a9z3 - 4a9z5 - a10z4

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 = 6                       1 j = 4                     2   j = 2                   5 1   j = 0                 7 2     j = -2               10 5       j = -4             10 8         j = -6           11 9           j = -8         8 11             j = -10       6 10               j = -12     3 8                 j = -14   1 6                   j = -16   3                     j = -18 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, 379]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 379]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[10, 15, 1, 16], > X[18, 6, 19, 5], X[6, 11, 7, 12], X[16, 7, 17, 8], X[22, 17, 11, 18], > X[4, 20, 5, 19], X[8, 22, 9, 21], X[20, 10, 21, 9]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -9, 5, -6, 7, -10, 11, -4}, > {6, -1, 2, -3, 4, -7, 8, -5, 9, -11, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 4 9 14 18 21 19 17 12 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 7 Sqrt[q] + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 -22 3 3 2 3 2 6 -6 3 2 -3 - q + q + q - --- + --- - --- + --- - --- + -- - q + -- + -- + 20 18 16 12 10 8 4 2 q q q q q q q q 2 4 8 > 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 379]][a, z]
Out[8]=	3 3 a a 2 z 3 5 7 z 3 3 3 -(-) + -- - --- + 2 a z - a z + a z - a z - -- + 5 a z - 3 a z + z z a a 5 3 7 3 5 3 5 5 5 3 7 > 4 a z - a z + 2 a z - 3 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 379]][a, z]
Out[9]=	3 2 a a 2 z 3 7 2 2 2 4 2 -a + - + -- + --- - a z - 5 a z + 2 a z + 5 z + 8 a z + 6 a z - z z a 3 8 2 5 z 3 3 3 5 3 7 3 9 3 4 > 3 a z - ---- - 2 a z + 3 a z - 11 a z - 10 a z + a z - 13 z - a 5 2 4 4 4 6 4 8 4 10 4 4 z 5 > 27 a z - 25 a z - 2 a z + 8 a z - a z + ---- - 6 a z - a 3 5 5 5 7 5 9 5 6 2 6 4 6 > 11 a z + 21 a z + 18 a z - 4 a z + 11 z + 26 a z + 39 a z + 7 6 6 8 6 z 7 3 7 7 7 8 2 8 > 15 a z - 9 a z - -- + 11 a z + 25 a z - 13 a z - 3 z - 4 a z - a 4 8 6 8 9 3 9 5 9 2 10 4 10 > 13 a z - 12 a z - 4 a z - 11 a z - 7 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 10 1 3 1 6 3 8 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 10 8 11 11 9 10 5 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 7 t + --- + 2 t + 5 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 2 3 4 3 6 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a379

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