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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 8q-15/2 + 13q-13/2 - 21q-11/2 + 21q-9/2 - 25q-7/2 + 20q-5/2 - 16q-3/2 + 10q-1/2 - 5q1/2 + q3/2

A2 (sl(3)) Invariant:

q-30 + q-28 + 6q-24 + 7q-22 + 6q-20 + 15q-18 + 9q-16 + 13q-14 + 11q-12 + 4q-10 + 8q-8 - 4q-6 + 3q-4 + q-2 - 2 + 3q2 - q4

HOMFLY-PT Polynomial:

- az + az3 + az5 - a3z-3 - 4a3z-1 - 4a3z - 4a3z3 - 3a3z5 - a3z7 + 3a5z-3 + 9a5z-1 + 10a5z + 8a5z3 + 3a5z5 - 3a7z-3 - 6a7z-1 - 6a7z - 3a7z3 + a9z-3 + a9z-1 + a9z

Kauffman Polynomial:

z4 - z6 - az - 4az3 + 10az5 - 5az7 - a2z2 - 10a2z4 + 20a2z6 - 9a2z8 + a3z-3 - 5a3z-1 + 11a3z - 14a3z3 + 13a3z5 + 6a3z7 - 7a3z9 - 3a4z-2 + 10a4 - 14a4z2 - 11a4z4 + 37a4z6 - 15a4z8 - 2a4z10 + 3a5z-3 - 12a5z-1 + 33a5z - 46a5z3 + 27a5z5 + 10a5z7 - 12a5z9 - 6a6z-2 + 19a6 - 26a6z2 + 25a6z6 - 13a6z8 - 2a6z10 + 3a7z-3 - 12a7z-1 + 33a7z - 50a7z3 + 35a7z5 - 7a7z7 - 5a7z9 - 3a8z-2 + 10a8 - 14a8z2 + 4a8z4 + 6a8z6 - 7a8z8 + a9z-3 - 5a9z-1 + 11a9z - 12a9z3 + 10a9z5 - 6a9z7 - a10z2 + 4a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

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 = 4                       1 j = 2                     4   j = 0                   6 1   j = -2                 10 4     j = -4               12 8       j = -6             13 8         j = -8           10 14           j = -10         11 11             j = -12       4 12               j = -14     4 9                 j = -16   1 6                   j = -18   2                     j = -20 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 545]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 545]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 5, 15, 6], X[12, 4, 13, 3], X[2, 9, 3, 10], > X[18, 7, 19, 8], X[8, 17, 9, 18], X[10, 13, 5, 14], X[22, 20, 17, 19], > X[16, 21, 11, 22], X[20, 11, 21, 12], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11}, {2, -1, 5, -6, 4, -7}, {10, -3, 7, -2, 11, -9}, > {6, -5, 8, -10, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 8 13 21 21 25 20 16 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 10 3/2 > ------- - 5 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 6 7 6 15 9 13 11 4 8 4 -2 + q + q + --- + --- + --- + --- + --- + --- + --- + --- + -- - -- + 24 22 20 18 16 14 12 10 8 6 q q q q q q q q q q 3 -2 2 4 > -- + q + 3 q - q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 545]][a, z]
Out[8]=	3 5 7 9 3 5 7 9 a 3 a 3 a a 4 a 9 a 6 a a 3 5 -(--) + ---- - ---- + -- - ---- + ---- - ---- + -- - a z - 4 a z + 10 a z - 3 3 3 3 z z z z z z z z 7 9 3 3 3 5 3 7 3 5 3 5 > 6 a z + a z + a z - 4 a z + 8 a z - 3 a z + a z - 3 a z + 5 5 3 7 > 3 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 545]][a, z]
Out[9]=	3 5 7 9 4 6 8 3 4 6 8 a 3 a 3 a a 3 a 6 a 3 a 5 a 10 a + 19 a + 10 a + -- + ---- + ---- + -- - ---- - ---- - ---- - ---- - 3 3 3 3 2 2 2 z z z z z z z z 5 7 9 12 a 12 a 5 a 3 5 7 9 > ----- - ----- - ---- - a z + 11 a z + 33 a z + 33 a z + 11 a z - z z z 11 2 2 4 2 6 2 8 2 10 2 3 > a z - a z - 14 a z - 26 a z - 14 a z - a z - 4 a z - 3 3 5 3 7 3 9 3 11 3 4 2 4 > 14 a z - 46 a z - 50 a z - 12 a z + 2 a z + z - 10 a z - 4 4 8 4 10 4 5 3 5 5 5 7 5 > 11 a z + 4 a z + 4 a z + 10 a z + 13 a z + 27 a z + 35 a z + 9 5 11 5 6 2 6 4 6 6 6 8 6 > 10 a z - a z - z + 20 a z + 37 a z + 25 a z + 6 a z - 10 6 7 3 7 5 7 7 7 9 7 2 8 > 3 a z - 5 a z + 6 a z + 10 a z - 7 a z - 6 a z - 9 a z - 4 8 6 8 8 8 3 9 5 9 7 9 4 10 > 15 a z - 13 a z - 7 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]=	8 10 1 2 1 6 4 9 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 12 11 11 10 14 13 8 12 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 6 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 4 t 2 2 2 4 3 > --- + t + 4 q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a545

L11a544 L11a546