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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 5q-7/2 + 8q-5/2 - 10q-3/2 + 12q-1/2 - 13q1/2 + 11q3/2 - 9q5/2 + 5q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + q-16 + 3q-12 - 2q-8 + q-6 - 3q-4 + 1 + 3q4 - q6 + 3q8 + 2q10 - q12 + 2q14 - q18

HOMFLY-PT Polynomial:

a-5z - a-3z-1 - 2a-3z - 2a-3z3 + a-1z-1 + 2a-1z + a-1z3 + a-1z5 + az-1 + az + az3 + az5 - 2a3z-1 - 3a3z - 2a3z3 + a5z-1 + a5z

Kauffman Polynomial:

a-6z2 - a-6z4 - 2a-5z + 4a-5z3 - 3a-5z5 + 3a-4z4 - 4a-4z6 + a-3z-1 - 5a-3z + 7a-3z3 - 4a-3z7 - a-2 + a-2z4 + 3a-2z6 - 4a-2z8 + a-1z-1 - 3a-1z + 2a-1z3 + 3a-1z7 - 3a-1z9 - 2 + 5z2 - 12z4 + 15z6 - 4z8 - z10 - az-1 + 8az - 13az3 + 12az7 - 5az9 - 3a2 + 8a2z2 - 18a2z4 + 16a2z6 - 2a2z8 - a2z10 - 2a3z-1 + 13a3z - 20a3z3 + 8a3z5 + 4a3z7 - 2a3z9 - a4 + 4a4z2 - 9a4z4 + 8a4z6 - 2a4z8 - a5z-1 + 5a5z - 8a5z3 + 5a5z5 - a5z7

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 = 12                       1 j = 10                     2   j = 8                   3 1   j = 6                 6 2     j = 4               5 3       j = 2             8 6         j = 0           6 7           j = -2         4 6             j = -4       4 6               j = -6     1 4                 j = -8   1 4                   j = -10   1                     j = -12 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, 43]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 43]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[18, 8, 19, 7], X[22, 13, 5, 14], > X[20, 15, 21, 16], X[16, 19, 17, 20], X[14, 21, 15, 22], X[12, 10, 13, 9], > X[8, 18, 9, 17], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 8, -2, 11, -8, 4, -7, 5, -6, 9, -3, > 6, -5, 7, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 5 8 10 12 3/2 -q + ---- - ---- + ---- - ---- + ------- - 13 Sqrt[q] + 11 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 9 q + 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 3 2 -6 3 4 6 8 10 12 1 + q + q + --- - -- + q - -- + 3 q - q + 3 q + 2 q - q + 12 8 4 q q q 14 18 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 43]][a, z]
Out[8]=	3 5 3 1 1 a 2 a a z 2 z 2 z 3 5 2 z -(----) + --- + - - ---- + -- + -- - --- + --- + a z - 3 a z + a z - ---- + 3 a z z z z 5 3 a 3 a z a a a 3 5 z 3 3 3 z 5 > -- + a z - 2 a z + -- + a z a a
In[9]:=	Kauffman[Link[11, Alternating, 43]][a, z]
Out[9]=	3 5 -2 2 4 1 1 a 2 a a 2 z 5 z 3 z -2 - a - 3 a - a + ---- + --- - - - ---- - -- - --- - --- - --- + 8 a z + 3 a z z z z 5 3 a a z a a 2 3 3 3 3 5 2 z 2 2 4 2 4 z 7 z 2 z > 13 a z + 5 a z + 5 z + -- + 8 a z + 4 a z + ---- + ---- + ---- - 6 5 3 a a a a 4 4 4 3 3 3 5 3 4 z 3 z z 2 4 > 13 a z - 20 a z - 8 a z - 12 z - -- + ---- + -- - 18 a z - 6 4 2 a a a 5 6 6 4 4 3 z 3 5 5 5 6 4 z 3 z 2 6 > 9 a z - ---- + 8 a z + 5 a z + 15 z - ---- + ---- + 16 a z + 5 4 2 a a a 7 7 8 4 6 4 z 3 z 7 3 7 5 7 8 4 z 2 8 > 8 a z - ---- + ---- + 12 a z + 4 a z - a z - 4 z - ---- - 2 a z - 3 a 2 a a 9 4 8 3 z 9 3 9 10 2 10 > 2 a z - ---- - 5 a z - 2 a z - z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 4 1 4 4 6 7 + 8 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 4 6 6 2 4 4 2 6 2 6 3 > ----- + - + ---- + 6 q t + 5 q t + 3 q t + 6 q t + 2 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a43

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