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

K11n82 K11n84

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X5,17,6,16 X7,12,8,13 X2,9,3,10 X11,18,12,19 X13,1,14,22 X15,21,16,20 X17,10,18,11 X19,7,20,6 X21,15,22,14

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 -16 -12 2 -18 -22 -20 -10 -6 -14

Alexander Polynomial:

3t-2 - 12t-1 + 19 - 12t + 3t2

Conway Polynomial:

1 + 3z4

Other knots with the same Alexander/Conway Polynomial:

{941, ...}

Determinant and Signature:

{49, 0}

Jones Polynomial:

- q-5 + 3q-4 - 5q-3 + 7q-2 - 8q-1 + 9 - 7q + 5q2 - 3q3 + q4

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

{927, ...}

A2 (sl(3)) Invariant:

- q-16 + q-14 + q-12 - 2q-10 + q-8 - q-6 + 3q-2 + 1 + 3q2 - 2q4 - q6 - 2q10 + q12 + q14

HOMFLY-PT Polynomial:

a-4 - 3a-2 - 3a-2z2 + 4 + 4z2 + 2z4 - a2 + a2z4 - a4z2

Kauffman Polynomial:

a-4 - 2a-4z2 + a-4z4 + a-3z - 5a-3z3 + 3a-3z5 + 3a-2 - 11a-2z2 + 10a-2z4 - 3a-2z6 + a-2z8 + a-1z - 7a-1z3 + 9a-1z5 - 3a-1z7 + a-1z9 + 4 - 15z2 + 23z4 - 13z6 + 4z8 - az + 6az3 - 5az5 + az7 + az9 + a2 - 4a2z2 + 7a2z4 - 7a2z6 + 3a2z8 - a3z + 6a3z3 - 10a3z5 + 4a3z7 + 2a4z2 - 7a4z4 + 3a4z6 - 2a5z3 + a5z5

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

{0, -1}

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=0 is the signature of 1183. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                   1 j = 7                 2   j = 5               3 1   j = 3             4 2     j = 1           5 3       j = -1         4 5         j = -3       3 4           j = -5     2 4             j = -7   1 3               j = -9   2                 j = -11 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, NonAlternating, 83]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 83]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[5, 17, 6, 16], X[7, 12, 8, 13], > X[2, 9, 3, 10], X[11, 18, 12, 19], X[13, 1, 14, 22], X[15, 21, 16, 20], > X[17, 10, 18, 11], X[19, 7, 20, 6], X[21, 15, 22, 14]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 83]]
Out[4]=	GaussCode[1, -5, 2, -1, -3, 10, -4, -2, 5, 9, -6, 4, -7, 11, -8, 3, -9, 6, -10, > 8, -11, 7]
In[5]:=	DTCode[Knot[11, NonAlternating, 83]]
Out[5]=	DTCode[4, 8, -16, -12, 2, -18, -22, -20, -10, -6, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 83]][t]
Out[6]=	3 12 2 19 + -- - -- - 12 t + 3 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 83]][z]
Out[7]=	4 1 + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 41], Knot[11, NonAlternating, 83]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 83]], KnotSignature[Knot[11, NonAlternating, 83]]}
Out[9]=	{49, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 83]][q]
Out[10]=	-5 3 5 7 8 2 3 4 9 - q + -- - -- + -- - - - 7 q + 5 q - 3 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 27], Knot[11, NonAlternating, 83]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 83]][q]
Out[12]=	-16 -14 -12 2 -8 -6 3 2 4 6 10 1 - q + q + q - --- + q - q + -- + 3 q - 2 q - q - 2 q + 10 2 q q 12 14 > q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 83]][a, z]
Out[13]=	2 -4 3 2 2 3 z 4 2 4 2 4 4 + a - -- - a + 4 z - ---- - a z + 2 z + a z 2 2 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 83]][a, z]
Out[14]=	2 2 -4 3 2 z z 3 2 2 z 11 z 2 2 4 + a + -- + a + -- + - - a z - a z - 15 z - ---- - ----- - 4 a z + 2 3 a 4 2 a a a a 3 3 4 4 4 2 5 z 7 z 3 3 3 5 3 4 z 10 z > 2 a z - ---- - ---- + 6 a z + 6 a z - 2 a z + 23 z + -- + ----- + 3 a 4 2 a a a 5 5 2 4 4 4 3 z 9 z 5 3 5 5 5 6 > 7 a z - 7 a z + ---- + ---- - 5 a z - 10 a z + a z - 13 z - 3 a a 6 7 8 3 z 2 6 4 6 3 z 7 3 7 8 z 2 8 > ---- - 7 a z + 3 a z - ---- + a z + 4 a z + 4 z + -- + 3 a z + 2 a 2 a a 9 z 9 > -- + a z a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 83]], Vassiliev[3][Knot[11, NonAlternating, 83]]}
Out[15]=	{0, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 83]][q, t]
Out[16]=	5 1 2 1 3 2 4 3 4 4 - + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 9 4 > 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n83

K11n82 K11n84