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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 6q-13/2 - 9q-11/2 + 12q-9/2 - 14q-7/2 + 13q-5/2 - 12q-3/2 + 8q-1/2 - 6q1/2 + 3q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-24 + q-22 - 2q-20 + q-18 - q-16 - q-14 + 2q-12 - 2q-10 + 5q-8 + 3q-4 + 2q-2 + 2q2 - q4 + q6

HOMFLY-PT Polynomial:

- 2az-1 - 6az - 8az3 - 5az5 - az7 + 3a3z-1 + 11a3z + 21a3z3 + 18a3z5 + 7a3z7 + a3z9 - a5z-1 - 5a5z - 8a5z3 - 5a5z5 - a5z7

Kauffman Polynomial:

a-1z - 4a-1z3 + 4a-1z5 - a-1z7 + 3z2 - 12z4 + 12z6 - 3z8 + 2az-1 - 7az + 12az3 - 19az5 + 16az7 - 4az9 - 3a2 + 10a2z2 - 20a2z4 + 13a2z6 + 2a2z8 - 2a2z10 + 3a3z-1 - 15a3z + 39a3z3 - 52a3z5 + 36a3z7 - 9a3z9 - 3a4 + 16a4z2 - 27a4z4 + 18a4z6 - a4z8 - 2a4z10 + a5z-1 - 6a5z + 14a5z3 - 17a5z5 + 13a5z7 - 5a5z9 - a6 + 6a6z2 - 12a6z4 + 12a6z6 - 6a6z8 + a7z - 6a7z3 + 9a7z5 - 6a7z7 - 2a8z2 + 6a8z4 - 5a8z6 + 3a9z3 - 3a9z5 + a10z2 - 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                   4 1   j = 0                 4 2     j = -2               8 4       j = -4             7 6         j = -6           7 6           j = -8         5 7             j = -10       4 7               j = -12     2 5                 j = -14   1 4                   j = -16   2                     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, 279]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 279]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[8, 9, 1, 10], > X[20, 13, 21, 14], X[14, 6, 15, 5], X[4, 20, 5, 19], X[18, 8, 19, 7], > X[6, 16, 7, 15], X[22, 17, 9, 18], X[16, 21, 17, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7, 6, -9, 8, -4}, > {4, -1, 2, -3, 5, -6, 9, -11, 10, -8, 7, -5, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 6 9 12 14 13 12 8 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 > 6 Sqrt[q] + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 2 -18 -16 -14 2 2 5 3 2 2 -q + q - --- + q - q - q + --- - --- + -- + -- + -- + 2 q - 20 12 10 8 4 2 q q q q q q 4 6 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 279]][a, z]
Out[8]=	3 5 -2 a 3 a a 3 5 3 3 3 5 3 ---- + ---- - -- - 6 a z + 11 a z - 5 a z - 8 a z + 21 a z - 8 a z - z z z 5 3 5 5 5 7 3 7 5 7 3 9 > 5 a z + 18 a z - 5 a z - a z + 7 a z - a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 279]][a, z]
Out[9]=	3 5 2 4 6 2 a 3 a a z 3 5 7 -3 a - 3 a - a + --- + ---- + -- + - - 7 a z - 15 a z - 6 a z + a z + z z z a 3 2 2 2 4 2 6 2 8 2 10 2 4 z 3 > 3 z + 10 a z + 16 a z + 6 a z - 2 a z + a z - ---- + 12 a z + a 3 3 5 3 7 3 9 3 4 2 4 4 4 > 39 a z + 14 a z - 6 a z + 3 a z - 12 z - 20 a z - 27 a z - 5 6 4 8 4 10 4 4 z 5 3 5 5 5 > 12 a z + 6 a z - a z + ---- - 19 a z - 52 a z - 17 a z + a 7 7 5 9 5 6 2 6 4 6 6 6 8 6 z > 9 a z - 3 a z + 12 z + 13 a z + 18 a z + 12 a z - 5 a z - -- + a 7 3 7 5 7 7 7 8 2 8 4 8 > 16 a z + 36 a z + 13 a z - 6 a z - 3 z + 2 a z - a z - 6 8 9 3 9 5 9 2 10 4 10 > 6 a z - 4 a z - 9 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 8 1 2 1 4 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 7 5 7 7 6 7 4 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 4 t + --- + 2 t + 4 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 L11a279

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