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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 6q-9/2 + 9q-7/2 - 11q-5/2 + 12q-3/2 - 12q-1/2 + 8q1/2 - 6q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

q-20 + 2q-14 - 2q-12 + q-10 - q-8 - 2q-6 + 2q-4 - q-2 + 5 + q2 + q4 + 2q6 - q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z-1 - 3a-1z - 3a-1z3 - a-1z5 + 4az-1 + 9az + 10az3 + 5az5 + az7 - 3a3z-1 - 8a3z - 7a3z3 - 2a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

2a-3z3 - a-3z5 - a-2z2 + 6a-2z4 - 3a-2z6 - 2a-1z-1 + 7a-1z - 13a-1z3 + 13a-1z5 - 5a-1z7 + 2 - z2 - 4z4 + 7z6 - 4z8 - 4az-1 + 19az - 40az3 + 31az5 - 9az7 - az9 + 3a2 - 5a2z2 - 7a2z4 + 13a2z6 - 7a2z8 - 3a3z-1 + 15a3z - 28a3z3 + 24a3z5 - 8a3z7 - a3z9 + 3a4 - 8a4z2 + 9a4z4 - 3a4z8 - a5z-1 + 2a5z - a5z3 + 6a5z5 - 4a5z7 + a6 - 3a6z2 + 6a6z4 - 3a6z6 - a7z + 2a7z3 - a7z5

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 j = 8                     1 j = 6                   2   j = 4                 4 1   j = 2               4 2     j = 0             8 4       j = -2           6 6         j = -4         5 6           j = -6       4 6             j = -8     2 5               j = -10   1 4                 j = -12   2                   j = -14 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[10, Alternating, 28]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 28]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[20, 16, 5, 15], > X[16, 9, 17, 10], X[8, 19, 9, 20], X[18, 11, 19, 12], X[10, 17, 11, 18], > X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -6, 5, -8, 7, -2, 10, -3, 4, -5, 8, -7, > 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 6 9 11 12 12 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 8 Sqrt[q] - 6 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 > 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 2 -10 -8 2 2 -2 2 4 6 8 10 5 + q + --- - --- + q - q - -- + -- - q + q + q + 2 q - q + q 14 12 6 4 q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 28]][a, z]
Out[8]=	3 5 3 -2 4 a 3 a a 3 z 3 5 3 z 3 --- + --- - ---- + -- - --- + 9 a z - 8 a z + 2 a z - ---- + 10 a z - a z z z z a a 5 3 3 5 3 z 5 3 5 7 > 7 a z + a z - -- + 5 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[10, Alternating, 28]][a, z]
Out[9]=	3 5 2 4 6 2 4 a 3 a a 7 z 3 2 + 3 a + 3 a + a - --- - --- - ---- - -- + --- + 19 a z + 15 a z + a z z z z a 2 3 3 5 7 2 z 2 2 4 2 6 2 2 z 13 z > 2 a z - a z - z - -- - 5 a z - 8 a z - 3 a z + ---- - ----- - 2 3 a a a 4 3 3 3 5 3 7 3 4 6 z 2 4 4 4 > 40 a z - 28 a z - a z + 2 a z - 4 z + ---- - 7 a z + 9 a z + 2 a 5 5 6 6 4 z 13 z 5 3 5 5 5 7 5 6 3 z > 6 a z - -- + ----- + 31 a z + 24 a z + 6 a z - a z + 7 z - ---- + 3 a 2 a a 7 2 6 6 6 5 z 7 3 7 5 7 8 2 8 > 13 a z - 3 a z - ---- - 9 a z - 8 a z - 4 a z - 4 z - 7 a z - a 4 8 9 3 9 > 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 2 1 4 2 5 4 6 8 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 5 6 6 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 4 t + 4 q t + 2 q t + 4 q t + q t + 2 q t + 4 2 4 2 q t q t q t 8 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a28

L10a27 L10a29