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

K11n118 K11n120

Knotscape

This page is passe. Go here instead!    The Knot K11n119 Visit K11n119's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X14,5,15,6 X7,20,8,21 X2,10,3,9 X16,11,17,12 X18,14,19,13 X8,15,9,16 X22,17,1,18 X19,6,20,7 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 6t-2 - 16t-1 + 23 - 16t + 6t2 - t3

Conway Polynomial:

1 - z2 - z6

Other knots with the same Alexander/Conway Polynomial:

{934, K11n32, ...}

Determinant and Signature:

{69, 0}

Jones Polynomial:

q-6 - 3q-5 + 6q-4 - 9q-3 + 11q-2 - 12q-1 + 11 - 8q + 6q2 - 2q3

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

{...}

A2 (sl(3)) Invariant:

q-18 - q-16 + 2q-14 + q-12 - 3q-10 + q-8 - 3q-6 + q-2 + 4q2 - q4 + 2q6 + q8 - 2q10

HOMFLY-PT Polynomial:

- a-2 - 2a-2z2 + 5 + 7z2 + 3z4 - 5a2 - 8a2z2 - 4a2z4 - a2z6 + 2a4 + 2a4z2 + a4z4

Kauffman Polynomial:

- a-3z + 3a-3z3 + a-2 - 7a-2z2 + 6a-2z4 + a-2z6 - 2a-1z + 6a-1z3 - 5a-1z5 + 4a-1z7 + 5 - 20z2 + 24z4 - 13z6 + 5z8 - 2az + 8az3 - 10az5 + 2az7 + 2az9 + 5a2 - 19a2z2 + 29a2z4 - 26a2z6 + 9a2z8 - 3a3z + 12a3z3 - 14a3z5 + a3z7 + 2a3z9 + 2a4 - 4a4z2 + 8a4z4 - 11a4z6 + 4a4z8 - 2a5z + 7a5z3 - 9a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

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

{-1, 2}

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 11119. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7                   2 j = 5                 4   j = 3               4 2   j = 1             7 4     j = -1           6 5       j = -3         5 6         j = -5       4 6           j = -7     2 5             j = -9   1 4               j = -11   2                 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, 119]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 119]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 5, 15, 6], X[7, 20, 8, 21], > X[2, 10, 3, 9], X[16, 11, 17, 12], X[18, 14, 19, 13], X[8, 15, 9, 16], > X[22, 17, 1, 18], X[19, 6, 20, 7], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 119]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, 10, -4, -8, 5, -2, 6, -11, 7, -3, 8, -6, 9, -7, -10, > 4, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 119]]
Out[5]=	DTCode[4, 10, 14, -20, 2, 16, 18, 8, 22, -6, 12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 119]][t]
Out[6]=	-3 6 16 2 3 23 - t + -- - -- - 16 t + 6 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 119]][z]
Out[7]=	2 6 1 - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 34], Knot[11, NonAlternating, 32], Knot[11, NonAlternating, 119]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 119]], KnotSignature[Knot[11, NonAlternating, 119]]}
Out[9]=	{69, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 119]][q]
Out[10]=	-6 3 6 9 11 12 2 3 11 + q - -- + -- - -- + -- - -- - 8 q + 6 q - 2 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, 119]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 119]][q]
Out[12]=	-18 -16 2 -12 3 -8 3 -2 2 4 6 8 10 q - q + --- + q - --- + q - -- + q + 4 q - q + 2 q + q - 2 q 14 10 6 q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 119]][a, z]
Out[13]=	2 -2 2 4 2 2 z 2 2 4 2 4 2 4 5 - a - 5 a + 2 a + 7 z - ---- - 8 a z + 2 a z + 3 z - 4 a z + 2 a 4 4 2 6 > a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 119]][a, z]
Out[14]=	2 -2 2 4 z 2 z 3 5 2 7 z 5 + a + 5 a + 2 a - -- - --- - 2 a z - 3 a z - 2 a z - 20 z - ---- - 3 a 2 a a 3 3 2 2 4 2 6 2 3 z 6 z 3 3 3 5 3 > 19 a z - 4 a z + 2 a z + ---- + ---- + 8 a z + 12 a z + 7 a z + 3 a a 4 5 4 6 z 2 4 4 4 6 4 5 z 5 3 5 > 24 z + ---- + 29 a z + 8 a z - 3 a z - ---- - 10 a z - 14 a z - 2 a a 6 7 5 5 6 z 2 6 4 6 6 6 4 z 7 > 9 a z - 13 z + -- - 26 a z - 11 a z + a z + ---- + 2 a z + 2 a a 3 7 5 7 8 2 8 4 8 9 3 9 > a z + 3 a z + 5 z + 9 a z + 4 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 119]], Vassiliev[3][Knot[11, NonAlternating, 119]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 119]][q, t]
Out[16]=	5 1 2 1 4 2 5 4 6 - + 7 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 5 6 6 3 3 2 5 2 7 3 > ----- + ---- + --- + 4 q t + 4 q t + 2 q t + 4 q t + 2 q t 3 2 3 q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n119

K11n118 K11n120