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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X10,5,11,6 X14,3,15,4 X18,13,19,14 X22,20,5,19 X20,12,21,11 X12,22,13,21 X2,9,3,10 X8,15,9,16

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 4q-15/2 - 8q-13/2 + 13q-11/2 - 18q-9/2 + 20q-7/2 - 22q-5/2 + 18q-3/2 - 14q-1/2 + 9q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-26 - 2q-24 - q-22 + 2q-20 - 4q-18 + 3q-16 + 3q-14 + 6q-10 + 4q-6 - 3q-2 + 3 - 4q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 + az-1 + 3az + az3 - az5 - 4a3z-1 - 10a3z - 8a3z3 - 3a3z5 + 4a5z-1 + 5a5z + a5z3 - a5z5 - a7z-1 + a7z3

Kauffman Polynomial:

- a-2z4 + a-1z3 - 4a-1z5 + 1 - 3z2 + 8z4 - 9z6 - az-1 + 4az - 11az3 + 18az5 - 13az7 + 4a2 - 9a2z2 + 5a2z4 + 13a2z6 - 12a2z8 - 4a3z-1 + 15a3z - 34a3z3 + 37a3z5 - 4a3z7 - 7a3z9 + 7a4 - 8a4z2 - 24a4z4 + 47a4z6 - 16a4z8 - 2a4z10 - 4a5z-1 + 14a5z - 27a5z3 + 8a5z5 + 23a5z7 - 12a5z9 + 4a6 + 2a6z2 - 35a6z4 + 39a6z6 - 8a6z8 - 2a6z10 - a7z-1 + 4a7z - 8a7z3 - 4a7z5 + 13a7z7 - 5a7z9 + a8 + 4a8z2 - 15a8z4 + 14a8z6 - 4a8z8 + a9z - 3a9z3 + 3a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                       1 j = 4                     3   j = 2                   6 1   j = 0                 8 3     j = -2               11 7       j = -4             11 7         j = -6           9 11           j = -8         9 11             j = -10       5 10               j = -12     3 8                 j = -14   1 5                   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, 134]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 134]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[10, 5, 11, 6], > X[14, 3, 15, 4], X[18, 13, 19, 14], X[22, 20, 5, 19], X[20, 12, 21, 11], > X[12, 22, 13, 21], X[2, 9, 3, 10], X[8, 15, 9, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {4, -1, 2, -11, 10, -4, 8, -9, 6, -5, 11, -2, 3, -6, > 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 4 8 13 18 20 22 18 14 -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 > 9 Sqrt[q] - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 2 -22 2 4 3 3 6 4 3 2 6 8 3 + q - --- - q + --- - --- + --- + --- + --- + -- - -- - 4 q + 2 q - q 24 20 18 16 14 10 6 2 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 134]][a, z]
Out[8]=	3 5 7 3 a 4 a 4 a a 3 5 z 3 3 3 5 3 - - ---- + ---- - -- + 3 a z - 10 a z + 5 a z + -- + a z - 8 a z + a z + z z z z a 7 3 5 3 5 5 5 > a z - a z - 3 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 134]][a, z]
Out[9]=	3 5 7 2 4 6 8 a 4 a 4 a a 3 1 + 4 a + 7 a + 4 a + a - - - ---- - ---- - -- + 4 a z + 15 a z + z z z z 5 7 9 2 2 2 4 2 6 2 8 2 > 14 a z + 4 a z + a z - 3 z - 9 a z - 8 a z + 2 a z + 4 a z + 3 4 z 3 3 3 5 3 7 3 9 3 4 z > -- - 11 a z - 34 a z - 27 a z - 8 a z - 3 a z + 8 z - -- + a 2 a 5 2 4 4 4 6 4 8 4 4 z 5 3 5 > 5 a z - 24 a z - 35 a z - 15 a z - ---- + 18 a z + 37 a z + a 5 5 7 5 9 5 6 2 6 4 6 6 6 > 8 a z - 4 a z + 3 a z - 9 z + 13 a z + 47 a z + 39 a z + 8 6 7 3 7 5 7 7 7 9 7 2 8 > 14 a z - 13 a z - 4 a z + 23 a z + 13 a z - a z - 12 a z - 4 8 6 8 8 8 3 9 5 9 7 9 4 10 > 16 a z - 8 a z - 4 a z - 7 a z - 12 a z - 5 a z - 2 a z - 6 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 3 1 5 3 8 5 8 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 10 9 11 9 11 11 7 11 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 3 t + 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q t q t q t q t q t q t q t q t 2 2 2 4 2 6 3 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a134

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