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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-21/2 - 5q-19/2 + 10q-17/2 - 16q-15/2 + 22q-13/2 - 25q-11/2 + 25q-9/2 - 22q-7/2 + 15q-5/2 - 10q-3/2 + 4q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-32 + 3q-30 + q-28 - 3q-26 + 5q-24 - 3q-22 - 2q-20 + 2q-18 - 4q-16 + 4q-14 - 2q-12 + 3q-10 + 5q-8 - 3q-6 + 5q-4 - 2 + q2

HOMFLY-PT Polynomial:

- az3 - 2a3z-1 - 5a3z - 2a3z3 + a3z5 + 4a5z-1 + 7a5z + 6a5z3 + 3a5z5 - 3a7z-1 - 5a7z - 2a7z3 + a7z5 + a9z-1 + a9z - a9z3

Kauffman Polynomial:

az3 - az5 + 4a2z4 - 4a2z6 - 2a3z-1 + 7a3z - 12a3z3 + 15a3z5 - 9a3z7 + 2a4 + a4z2 - 11a4z4 + 17a4z6 - 11a4z8 - 4a5z-1 + 16a5z - 36a5z3 + 33a5z5 - 5a5z7 - 7a5z9 + 3a6 + 5a6z2 - 38a6z4 + 52a6z6 - 21a6z8 - 2a6z10 - 3a7z-1 + 13a7z - 33a7z3 + 24a7z5 + 13a7z7 - 14a7z9 + 3a8 + 5a8z2 - 36a8z4 + 52a8z6 - 19a8z8 - 2a8z10 - a9z-1 + 3a9z - 14a9z3 + 17a9z5 + 4a9z7 - 7a9z9 + a10 + a10z2 - 12a10z4 + 20a10z6 - 9a10z8 - a11z - 4a11z3 + 10a11z5 - 5a11z7 + a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                       1 j = 0                     3   j = -2                   7 1   j = -4                 9 4     j = -6               13 6       j = -8             12 9         j = -10           13 13           j = -12         10 13             j = -14       6 12               j = -16     4 10                 j = -18   1 6                   j = -20   4                     j = -22 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, 133]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 133]]
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, 11, 19, 12], X[22, 19, 5, 20], X[20, 14, 21, 13], > 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, 6, -9, 8, -5, 11, -2, 3, -6, > 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 5 10 16 22 25 25 22 15 10 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 4 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 3 -28 3 5 3 2 2 4 4 2 3 -2 - q + --- + q - --- + --- - --- - --- + --- - --- + --- - --- + --- + 30 26 24 22 20 18 16 14 12 10 q q q q q q q q q q 5 3 5 2 > -- - -- + -- + q 8 6 4 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 133]][a, z]
Out[8]=	3 5 7 9 -2 a 4 a 3 a a 3 5 7 9 3 3 3 ----- + ---- - ---- + -- - 5 a z + 7 a z - 5 a z + a z - a z - 2 a z + z z z z 5 3 7 3 9 3 3 5 5 5 7 5 > 6 a z - 2 a z - a z + a z + 3 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 133]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 2 a 4 a 3 a a 3 5 2 a + 3 a + 3 a + a - ---- - ---- - ---- - -- + 7 a z + 16 a z + z z z z 7 9 11 4 2 6 2 8 2 10 2 3 > 13 a z + 3 a z - a z + a z + 5 a z + 5 a z + a z + a z - 3 3 5 3 7 3 9 3 11 3 2 4 4 4 > 12 a z - 36 a z - 33 a z - 14 a z - 4 a z + 4 a z - 11 a z - 6 4 8 4 10 4 12 4 5 3 5 5 5 > 38 a z - 36 a z - 12 a z + a z - a z + 15 a z + 33 a z + 7 5 9 5 11 5 2 6 4 6 6 6 > 24 a z + 17 a z + 10 a z - 4 a z + 17 a z + 52 a z + 8 6 10 6 12 6 3 7 5 7 7 7 9 7 > 52 a z + 20 a z - a z - 9 a z - 5 a z + 13 a z + 4 a z - 11 7 4 8 6 8 8 8 10 8 5 9 7 9 > 5 a z - 11 a z - 21 a z - 19 a z - 9 a z - 7 a z - 14 a z - 9 9 6 10 8 10 > 7 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 7 1 4 1 6 4 10 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 12 10 13 13 13 12 9 13 6 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 9 t 2 2 > ---- + 3 t + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a133

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