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

K11n100 K11n102

Knotscape

This page is passe. Go here instead!    The Knot K11n101 Visit K11n101's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,14,6,15 X7,12,8,13 X18,9,19,10 X2,11,3,12 X13,6,14,7 X22,16,1,15 X20,18,21,17 X8,19,9,20 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 2t-2 + 10t-1 - 15 + 10t - 2t2

Conway Polynomial:

1 + 2z2 - 2z4

Other knots with the same Alexander/Conway Polynomial:

{915, 10165, K11n63, ...}

Determinant and Signature:

{39, -2}

Jones Polynomial:

- q-6 + 2q-5 - 4q-4 + 6q-3 - 6q-2 + 7q-1 - 5 + 4q - 3q2 + q3

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

{...}

A2 (sl(3)) Invariant:

- q-20 - q-18 + q-16 - q-14 + 2q-10 + 2q-6 + q-2 + 1 - q2 + q4 - q6 - q8 + q10

HOMFLY-PT Polynomial:

a-2z2 - z2 - z4 + a2 - a2z4 + a4 + 2a4z2 - a6

Kauffman Polynomial:

a-2z2 - 3a-2z4 + a-2z6 + 8a-1z3 - 11a-1z5 + 3a-1z7 - 3z2 + 10z4 - 11z6 + 3z8 + az + 6az3 - 8az5 + az9 - a2 - 8a2z2 + 19a2z4 - 15a2z6 + 4a2z8 + 3a3z - 6a3z3 + 6a3z5 - 3a3z7 + a3z9 + a4 - 6a4z2 + 8a4z4 - 3a4z6 + a4z8 + a5z - 3a5z3 + 3a5z5 + a6 - 2a6z2 + 2a6z4 - a7z + a7z3

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

{2, -3}

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 11101. 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 = 7                   1 j = 5                 2   j = 3               2 1   j = 1             3 2     j = -1           4 2       j = -3         3 4         j = -5       3 3           j = -7     1 3             j = -9   1 3               j = -11   1                 j = -13 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, 101]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 101]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 14, 6, 15], X[7, 12, 8, 13], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[13, 6, 14, 7], X[22, 16, 1, 15], > X[20, 18, 21, 17], X[8, 19, 9, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 101]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 7, -4, -10, 5, -2, 6, 4, -7, 3, 8, -11, 9, -5, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 101]]
Out[5]=	DTCode[4, 10, -14, -12, 18, 2, -6, 22, 20, 8, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 101]][t]
Out[6]=	2 10 2 -15 - -- + -- + 10 t - 2 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 101]][z]
Out[7]=	2 4 1 + 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 15], Knot[10, 165], Knot[11, NonAlternating, 63], > Knot[11, NonAlternating, 101]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 101]], KnotSignature[Knot[11, NonAlternating, 101]]}
Out[9]=	{39, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 101]][q]
Out[10]=	-6 2 4 6 6 7 2 3 -5 - q + -- - -- + -- - -- + - + 4 q - 3 q + q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 101]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 101]][q]
Out[12]=	-20 -18 -16 -14 2 2 -2 2 4 6 8 10 1 - q - q + q - q + --- + -- + q - q + q - q - q + q 10 6 q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 101]][a, z]
Out[13]=	2 2 4 6 2 z 4 2 4 2 4 a + a - a - z + -- + 2 a z - z - a z 2 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 101]][a, z]
Out[14]=	2 2 4 6 3 5 7 2 z 2 2 4 2 -a + a + a + a z + 3 a z + a z - a z - 3 z + -- - 8 a z - 6 a z - 2 a 3 4 6 2 8 z 3 3 3 5 3 7 3 4 3 z > 2 a z + ---- + 6 a z - 6 a z - 3 a z + a z + 10 z - ---- + a 2 a 5 2 4 4 4 6 4 11 z 5 3 5 5 5 6 > 19 a z + 8 a z + 2 a z - ----- - 8 a z + 6 a z + 3 a z - 11 z + a 6 7 z 2 6 4 6 3 z 3 7 8 2 8 4 8 9 > -- - 15 a z - 3 a z + ---- - 3 a z + 3 z + 4 a z + a z + a z + 2 a a 3 9 > a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 101]], Vassiliev[3][Knot[11, NonAlternating, 101]]}
Out[15]=	{2, -3}
In[16]:=	Kh[Knot[11, NonAlternating, 101]][q, t]
Out[16]=	4 4 1 1 1 3 1 3 3 3 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 3 2 t 2 3 2 3 3 5 3 7 4 > ---- + --- + 3 q t + 2 q t + 2 q t + q t + 2 q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n101

K11n100 K11n102