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

K11n84 K11n86

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X7,15,8,14 X16,10,17,9 X2,11,3,12 X13,19,14,18 X20,16,21,15 X22,17,1,18 X19,6,20,7 X8,21,9,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 5t-2 - 10t-1 + 13 - 10t + 5t2 - t3

Conway Polynomial:

1 + z2 - z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{818, 924, K11n164, ...}

Determinant and Signature:

{45, 0}

Jones Polynomial:

- q-3 + 3q-2 - 5q-1 + 7 - 7q + 8q2 - 6q3 + 4q4 - 3q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-10 + q-6 - q-4 + 2q-2 + q2 + 2q4 + 2q8 - 2q10 - q12 - q16 + q18

HOMFLY-PT Polynomial:

2a-4z2 + a-4z4 - a-2 - 5a-2z2 - 4a-2z4 - a-2z6 + 3 + 5z2 + 2z4 - a2 - a2z2

Kauffman Polynomial:

a-6z2 - 3a-6z4 + a-6z6 - a-5z + 9a-5z3 - 11a-5z5 + 3a-5z7 - 3a-4z2 + 8a-4z4 - 10a-4z6 + 3a-4z8 - 4a-3z + 17a-3z3 - 16a-3z5 + 2a-3z7 + a-3z9 + a-2 - 12a-2z2 + 23a-2z4 - 18a-2z6 + 5a-2z8 - 5a-1z + 11a-1z3 - 6a-1z5 + a-1z9 + 3 - 11z2 + 15z4 - 7z6 + 2z8 - 3az + 4az3 - az5 + az7 + a2 - 3a2z2 + 3a2z4 - a3z + a3z3

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

{1, 0}

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 1185. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                   1 j = 11                 2   j = 9               2 1   j = 7             4 2     j = 5           4 2       j = 3         3 4         j = 1       4 4           j = -1     2 4             j = -3   1 3               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, NonAlternating, 85]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 85]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[7, 15, 8, 14], > X[16, 10, 17, 9], X[2, 11, 3, 12], X[13, 19, 14, 18], X[20, 16, 21, 15], > X[22, 17, 1, 18], X[19, 6, 20, 7], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 85]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, 10, -4, -11, 5, -2, 6, -3, -7, 4, 8, -5, 9, 7, -10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 85]]
Out[5]=	DTCode[4, 10, 12, -14, 16, 2, -18, 20, 22, -6, 8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 85]][t]
Out[6]=	-3 5 10 2 3 13 - t + -- - -- - 10 t + 5 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 85]][z]
Out[7]=	2 4 6 1 + z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 18], Knot[9, 24], Knot[11, NonAlternating, 85], > Knot[11, NonAlternating, 164]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 85]], KnotSignature[Knot[11, NonAlternating, 85]]}
Out[9]=	{45, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 85]][q]
Out[10]=	-3 3 5 2 3 4 5 6 7 - q + -- - - - 7 q + 8 q - 6 q + 4 q - 3 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 85]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 85]][q]
Out[12]=	-10 -6 -4 2 2 4 8 10 12 16 18 -q + q - q + -- + q + 2 q + 2 q - 2 q - q - q + q 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 85]][a, z]
Out[13]=	2 2 4 4 6 -2 2 2 2 z 5 z 2 2 4 z 4 z z 3 - a - a + 5 z + ---- - ---- - a z + 2 z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 85]][a, z]
Out[14]=	2 2 2 -2 2 z 4 z 5 z 3 2 z 3 z 12 z 3 + a + a - -- - --- - --- - 3 a z - a z - 11 z + -- - ---- - ----- - 5 3 a 6 4 2 a a a a a 3 3 3 4 4 2 2 9 z 17 z 11 z 3 3 3 4 3 z 8 z > 3 a z + ---- + ----- + ----- + 4 a z + a z + 15 z - ---- + ---- + 5 3 a 6 4 a a a a 4 5 5 5 6 6 6 23 z 2 4 11 z 16 z 6 z 5 6 z 10 z 18 z > ----- + 3 a z - ----- - ----- - ---- - a z - 7 z + -- - ----- - ----- + 2 5 3 a 6 4 2 a a a a a a 7 7 8 8 9 9 3 z 2 z 7 8 3 z 5 z z z > ---- + ---- + a z + 2 z + ---- + ---- + -- + -- 5 3 4 2 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 85]], Vassiliev[3][Knot[11, NonAlternating, 85]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 85]][q, t]
Out[16]=	4 1 2 1 3 2 3 3 2 - + 4 q + ----- + ----- + ----- + ---- + --- + 4 q t + 3 q t + 4 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 13 6 > 4 q t + 2 q t + 4 q t + 2 q t + 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n85

K11n84 K11n86