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

K11a106 K11a108

Knotscape

This page is passe. Go here instead!    The Knot K11a107 Visit K11a107's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 16 2 20 18 8 6 22 12

Alexander Polynomial:

2t-3 - 11t-2 + 26t-1 - 33 + 26t - 11t2 + 2t3

Conway Polynomial:

1 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{10113, K11a347, ...}

Determinant and Signature:

{111, 2}

Jones Polynomial:

- q-4 + 3q-3 - 6q-2 + 11q-1 - 14 + 17q - 18q2 + 16q3 - 12q4 + 8q5 - 4q6 + q7

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

{...}

A2 (sl(3)) Invariant:

- q-12 + q-10 - q-8 - q-6 + 4q-4 - q-2 + 3 + 2q2 - 3q4 + q6 - 4q8 + 2q10 + q12 - q14 + 3q16 - 2q18 - q20 + q22

HOMFLY-PT Polynomial:

a-6z2 + a-4 - 2a-4z2 - 2a-4z4 - 3a-2 - 2a-2z2 + a-2z4 + a-2z6 + 4 + 5z2 + 3z4 + z6 - a2 - 2a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - 2a-7z3 + 4a-7z5 + 3a-6z2 - 8a-6z4 + 8a-6z6 - 2a-5z + 6a-5z3 - 12a-5z5 + 10a-5z7 + a-4 + 3a-4z2 - 9a-4z4 - 4a-4z6 + 8a-4z8 - 6a-3z + 26a-3z3 - 35a-3z5 + 9a-3z7 + 4a-3z9 + 3a-2 - 10a-2z2 + 24a-2z4 - 35a-2z6 + 13a-2z8 + a-2z10 - 8a-1z + 29a-1z3 - 21a-1z5 - 8a-1z7 + 7a-1z9 + 4 - 18z2 + 40z4 - 35z6 + 8z8 + z10 - 6az + 16az3 - 6az5 - 6az7 + 3az9 + a2 - 8a2z2 + 16a2z4 - 12a2z6 + 3a2z8 - 2a3z + 5a3z3 - 4a3z5 + a3z7

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=2 is the signature of 11107. 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 r = 5 r = 6 j = 15                       1 j = 13                     3   j = 11                   5 1   j = 9                 7 3     j = 7               9 5       j = 5             9 7         j = 3           8 9           j = 1         7 10             j = -1       4 7               j = -3     2 7                 j = -5   1 4                   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, 107]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 107]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 5, 15, 6], X[16, 7, 17, 8], > X[2, 10, 3, 9], X[20, 12, 21, 11], X[18, 13, 19, 14], X[8, 15, 9, 16], > X[6, 17, 7, 18], X[22, 20, 1, 19], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 107]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -8, 5, -2, 6, -11, 7, -3, 8, -4, 9, -7, 10, > -6, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 107]]
Out[5]=	DTCode[4, 10, 14, 16, 2, 20, 18, 8, 6, 22, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 107]][t]
Out[6]=	2 11 26 2 3 -33 + -- - -- + -- + 26 t - 11 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 107]][z]
Out[7]=	4 6 1 + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 113], Knot[11, Alternating, 107], Knot[11, Alternating, 347]}
In[9]:=	{KnotDet[Knot[11, Alternating, 107]], KnotSignature[Knot[11, Alternating, 107]]}
Out[9]=	{111, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 107]][q]
Out[10]=	-4 3 6 11 2 3 4 5 6 7 -14 - q + -- - -- + -- + 17 q - 18 q + 16 q - 12 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, 107]}
In[12]:=	A2Invariant[Knot[11, Alternating, 107]][q]
Out[12]=	-12 -10 -8 -6 4 -2 2 4 6 8 10 3 - q + q - q - q + -- - q + 2 q - 3 q + q - 4 q + 2 q + 4 q 12 14 16 18 20 22 > q - q + 3 q - 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 107]][a, z]
Out[13]=	2 2 2 4 4 -4 3 2 2 z 2 z 2 z 2 2 4 2 z z 4 + a - -- - a + 5 z + -- - ---- - ---- - 2 a z + 3 z - ---- + -- - 2 6 4 2 4 2 a a a a a a 6 2 4 6 z > a z + z + -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 107]][a, z]
Out[14]=	2 2 -4 3 2 2 z 6 z 8 z 3 2 3 z 3 z 4 + a + -- + a - --- - --- - --- - 6 a z - 2 a z - 18 z + ---- + ---- - 2 5 3 a 6 4 a a a a a 2 3 3 3 3 10 z 2 2 2 z 6 z 26 z 29 z 3 3 3 4 > ----- - 8 a z - ---- + ---- + ----- + ----- + 16 a z + 5 a z + 40 z + 2 7 5 3 a a a a a 4 4 4 4 5 5 5 5 z 8 z 9 z 24 z 2 4 4 z 12 z 35 z 21 z > -- - ---- - ---- + ----- + 16 a z + ---- - ----- - ----- - ----- - 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 7 7 5 3 5 6 8 z 4 z 35 z 2 6 10 z 9 z > 6 a z - 4 a z - 35 z + ---- - ---- - ----- - 12 a z + ----- + ---- - 6 4 2 5 3 a a a a a 7 8 8 9 9 8 z 7 3 7 8 8 z 13 z 2 8 4 z 7 z > ---- - 6 a z + a z + 8 z + ---- + ----- + 3 a z + ---- + ---- + a 4 2 3 a a a a 10 9 10 z > 3 a z + z + --- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 107]], Vassiliev[3][Knot[11, Alternating, 107]]}
Out[15]=	{0, -1}
In[16]:=	Kh[Knot[11, Alternating, 107]][q, t]
Out[16]=	3 1 2 1 4 2 7 4 7 10 q + 8 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 7 q 3 5 5 2 7 2 7 3 9 3 9 4 > --- + 9 q t + 9 q t + 7 q t + 9 q t + 5 q t + 7 q t + 3 q t + t 11 4 11 5 13 5 15 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a107

K11a106 K11a108