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

K11a114 K11a116

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X14,6,15,5 X18,7,19,8 X2,10,3,9 X20,11,21,12 X6,14,7,13 X22,16,1,15 X12,17,13,18 X8,19,9,20 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 18 2 20 6 22 12 8 16

Alexander Polynomial:

- 3t-3 + 13t-2 - 27t-1 + 35 - 27t + 13t2 - 3t3

Conway Polynomial:

1 - 2z2 - 5z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{121, 0}

Jones Polynomial:

q-4 - 3q-3 + 7q-2 - 12q-1 + 17 - 19q + 19q2 - 17q3 + 13q4 - 8q5 + 4q6 - q7

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

{...}

A2 (sl(3)) Invariant:

q-12 - q-10 + q-8 + 2q-6 - 3q-4 + 4q-2 - 1 - q2 + q4 - 4q6 + 3q8 - 2q10 + q12 + 3q14 - 2q16 + 2q18 - q22

HOMFLY-PT Polynomial:

- a-6 - a-6z2 + 4a-4 + 7a-4z2 + 3a-4z4 - 4a-2 - 9a-2z2 - 7a-2z4 - 2a-2z6 + 1 - z2 - 2z4 - z6 + a2 + 2a2z2 + a2z4

Kauffman Polynomial:

- a-7z + 3a-7z3 - 3a-7z5 + a-7z7 + a-6 - 5a-6z2 + 15a-6z4 - 14a-6z6 + 4a-6z8 - 3a-5z + 8a-5z3 + 3a-5z5 - 13a-5z7 + 5a-5z9 + 4a-4 - 23a-4z2 + 51a-4z4 - 44a-4z6 + 8a-4z8 + 2a-4z10 - 3a-3z + 6a-3z3 + 10a-3z5 - 29a-3z7 + 12a-3z9 + 4a-2 - 25a-2z2 + 54a-2z4 - 53a-2z6 + 14a-2z8 + 2a-2z10 - 3a-1z + 13a-1z3 - 12a-1z5 - 6a-1z7 + 7a-1z9 + 1 - 2z2 + 11z4 - 17z6 + 10z8 - 2az + 10az3 - 13az5 + 9az7 - a2 + 4a2z2 - 6a2z4 + 6a2z6 - 2a3z3 + 3a3z5 - a4z2 + a4z4

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

{-2, 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 11115. 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 = 15                       1 j = 13                     3   j = 11                   5 1   j = 9                 8 3     j = 7               9 5       j = 5             10 8         j = 3           9 9           j = 1         8 10             j = -1       5 10               j = -3     2 7                 j = -5   1 5                   j = -7   2                     j = -9 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, 115]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 115]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[18, 7, 19, 8], > X[2, 10, 3, 9], X[20, 11, 21, 12], X[6, 14, 7, 13], X[22, 16, 1, 15], > X[12, 17, 13, 18], X[8, 19, 9, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 115]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -7, 4, -10, 5, -2, 6, -9, 7, -3, 8, -11, 9, -4, 10, > -6, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 115]]
Out[5]=	DTCode[4, 10, 14, 18, 2, 20, 6, 22, 12, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 115]][t]
Out[6]=	3 13 27 2 3 35 - -- + -- - -- - 27 t + 13 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 115]][z]
Out[7]=	2 4 6 1 - 2 z - 5 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 115]}
In[9]:=	{KnotDet[Knot[11, Alternating, 115]], KnotSignature[Knot[11, Alternating, 115]]}
Out[9]=	{121, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 115]][q]
Out[10]=	-4 3 7 12 2 3 4 5 6 7 17 + q - -- + -- - -- - 19 q + 19 q - 17 q + 13 q - 8 q + 4 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 115]}
In[12]:=	A2Invariant[Knot[11, Alternating, 115]][q]
Out[12]=	-12 -10 -8 2 3 4 2 4 6 8 10 12 -1 + q - q + q + -- - -- + -- - q + q - 4 q + 3 q - 2 q + q + 6 4 2 q q q 14 16 18 22 > 3 q - 2 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 115]][a, z]
Out[13]=	2 2 2 4 4 -6 4 4 2 2 z 7 z 9 z 2 2 4 3 z 7 z 1 - a + -- - -- + a - z - -- + ---- - ---- + 2 a z - 2 z + ---- - ---- + 4 2 6 4 2 4 2 a a a a a a a 6 2 4 6 2 z > a z - z - ---- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 115]][a, z]
Out[14]=	2 2 -6 4 4 2 z 3 z 3 z 3 z 2 5 z 23 z 1 + a + -- + -- - a - -- - --- - --- - --- - 2 a z - 2 z - ---- - ----- - 4 2 7 5 3 a 6 4 a a a a a a a 2 3 3 3 3 25 z 2 2 4 2 3 z 8 z 6 z 13 z 3 3 3 > ----- + 4 a z - a z + ---- + ---- + ---- + ----- + 10 a z - 2 a z + 2 7 5 3 a a a a a 4 4 4 5 5 5 4 15 z 51 z 54 z 2 4 4 4 3 z 3 z 10 z > 11 z + ----- + ----- + ----- - 6 a z + a z - ---- + ---- + ----- - 6 4 2 7 5 3 a a a a a a 5 6 6 6 7 12 z 5 3 5 6 14 z 44 z 53 z 2 6 z > ----- - 13 a z + 3 a z - 17 z - ----- - ----- - ----- + 6 a z + -- - a 6 4 2 7 a a a a 7 7 7 8 8 8 9 13 z 29 z 6 z 7 8 4 z 8 z 14 z 5 z > ----- - ----- - ---- + 9 a z + 10 z + ---- + ---- + ----- + ---- + 5 3 a 6 4 2 5 a a a a a a 9 9 10 10 12 z 7 z 2 z 2 z > ----- + ---- + ----- + ----- 3 a 4 2 a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 115]], Vassiliev[3][Knot[11, Alternating, 115]]}
Out[15]=	{-2, 0}
In[16]:=	Kh[Knot[11, Alternating, 115]][q, t]
Out[16]=	10 1 2 1 5 2 7 5 -- + 8 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 10 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 9 q t + 9 q t + 10 q t + 8 q t + 9 q t + 5 q t + 8 q t + 9 5 11 5 11 6 13 6 15 7 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a115

K11a114 K11a116