© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a16 K11a18

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X12,6,13,5 X2837 X16,9,17,10 X6,12,7,11 X20,14,21,13 X10,15,11,16 X22,17,1,18 X14,20,15,19 X18,21,19,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 16 6 20 10 22 14 18

Alexander Polynomial:

t-3 - 10t-2 + 30t-1 - 41 + 30t - 10t2 + t3

Conway Polynomial:

1 - z2 - 4z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{123, 2}

Jones Polynomial:

q-3 - 3q-2 + 8q-1 - 13 + 17q - 20q2 + 20q3 - 17q4 + 13q5 - 7q6 + 3q7 - q8

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 4q-4 - 2q-2 + 3q2 - 5q4 + 2q6 - 3q8 + q10 + 2q12 - 2q14 + 5q16 - q18 - q20 + 2q22 - q24 - q26

HOMFLY-PT Polynomial:

- a-8 + 2a-6 + 3a-6z2 - 3a-4z2 - 3a-4z4 - a-2 + a-2z4 + a-2z6 - 2z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

a-9z - 2a-9z3 + a-9z5 - a-8 + 3a-8z2 - 5a-8z4 + 3a-8z6 + 2a-7z3 - 6a-7z5 + 5a-7z7 - 2a-6 + 3a-6z2 + a-6z4 - 6a-6z6 + 6a-6z8 - 2a-5z + 13a-5z3 - 16a-5z5 + 4a-5z7 + 4a-5z9 - 8a-4z2 + 26a-4z4 - 32a-4z6 + 14a-4z8 + a-4z10 + 10a-3z3 - 13a-3z5 - 5a-3z7 + 8a-3z9 + a-2 - 12a-2z2 + 28a-2z4 - 34a-2z6 + 13a-2z8 + a-2z10 + 6a-1z3 - 11a-1z5 - a-1z7 + 4a-1z9 - z2 + 5z4 - 10z6 + 5z8 - az + 5az3 - 7az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-1, 2}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=2 is the signature of 1117. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     2   j = 13                   5 1   j = 11                 8 2     j = 9               9 5       j = 7             11 8         j = 5           9 9           j = 3         8 11             j = 1       6 10               j = -1     2 7                 j = -3   1 6                   j = -5   2                     j = -7 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]:=	Show[DrawMorseLink[Knot[11, Alternating, 17]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 17]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[2, 8, 3, 7], > X[16, 9, 17, 10], X[6, 12, 7, 11], X[20, 14, 21, 13], X[10, 15, 11, 16], > X[22, 17, 1, 18], X[14, 20, 15, 19], X[18, 21, 19, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 17]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -6, 4, -2, 5, -8, 6, -3, 7, -10, 8, -5, 9, -11, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 17]]
Out[5]=	DTCode[4, 8, 12, 2, 16, 6, 20, 10, 22, 14, 18]
In[6]:=	alex = Alexander[Knot[11, Alternating, 17]][t]
Out[6]=	-3 10 30 2 3 -41 + t - -- + -- + 30 t - 10 t + t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 17]][z]
Out[7]=	2 4 6 1 - z - 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 17]}
In[9]:=	{KnotDet[Knot[11, Alternating, 17]], KnotSignature[Knot[11, Alternating, 17]]}
Out[9]=	{123, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 17]][q]
Out[10]=	-3 3 8 2 3 4 5 6 7 8 -13 + q - -- + - + 17 q - 20 q + 20 q - 17 q + 13 q - 7 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 17]}
In[12]:=	A2Invariant[Knot[11, Alternating, 17]][q]
Out[12]=	-10 -6 4 2 2 4 6 8 10 12 14 q - q + -- - -- + 3 q - 5 q + 2 q - 3 q + q + 2 q - 2 q + 4 2 q q 16 18 20 22 24 26 > 5 q - q - q + 2 q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 17]][a, z]
Out[13]=	2 2 4 4 6 -8 2 -2 2 2 3 z 3 z 2 2 4 3 z z z -a + -- - a + a - 2 z + ---- - ---- + a z - 2 z - ---- + -- + -- 6 6 4 4 2 2 a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 17]][a, z]
Out[14]=	2 2 2 2 -8 2 -2 2 z 2 z 2 3 z 3 z 8 z 12 z -a - -- + a - a + -- - --- - a z - z + ---- + ---- - ---- - ----- + 6 9 5 8 6 4 2 a a a a a a a 3 3 3 3 3 4 4 2 2 2 z 2 z 13 z 10 z 6 z 3 4 5 z z > 3 a z - ---- + ---- + ----- + ----- + ---- + 5 a z + 5 z - ---- + -- + 9 7 5 3 a 8 6 a a a a a a 4 4 5 5 5 5 5 26 z 28 z 2 4 z 6 z 16 z 13 z 11 z 5 > ----- + ----- - 3 a z + -- - ---- - ----- - ----- - ----- - 7 a z - 4 2 9 7 5 3 a a a a a a a 6 6 6 6 7 7 7 7 6 3 z 6 z 32 z 34 z 2 6 5 z 4 z 5 z z > 10 z + ---- - ---- - ----- - ----- + a z + ---- + ---- - ---- - -- + 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 7 8 6 z 14 z 13 z 4 z 8 z 4 z z z > 3 a z + 5 z + ---- + ----- + ----- + ---- + ---- + ---- + --- + --- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 17]], Vassiliev[3][Knot[11, Alternating, 17]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, Alternating, 17]][q, t]
Out[16]=	3 1 2 1 6 2 7 6 q 3 10 q + 8 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 11 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 9 q t + 9 q t + 11 q t + 8 q t + 9 q t + 5 q t + 8 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a17

K11a16 K11a18