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

K11n115 K11n117

Knotscape

This page is passe. Go here instead!    The Knot K11n116 Visit K11n116's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,14,6,15 X7,19,8,18 X9,17,10,16 X2,11,3,12 X20,13,21,14 X15,22,16,1 X17,9,18,8 X12,19,13,20 X21,7,22,6

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-2 + 3 - t2

Conway Polynomial:

1 - 4z2 - z4

Other knots with the same Alexander/Conway Polynomial:

{K11n49, ...}

Determinant and Signature:

{1, 0}

Jones Polynomial:

q-6 - q-5 + q-4 - q-3 + q - q2 + q3

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

{...}

A2 (sl(3)) Invariant:

q-20 + q-18 - q-12 - q-2 + q6 + q8 + q10

HOMFLY-PT Polynomial:

2a-2 + a-2z2 - 2 - 4z2 - z4 + a2 - a4 - a4z2 + a6

Kauffman Polynomial:

- 2a-2 + 6a-2z2 - 5a-2z4 + a-2z6 + 5a-1z3 - 5a-1z5 + a-1z7 - 2 + 8z2 - 6z4 + z6 + 2az - 2az3 - a2 + a2z2 - a2z4 - a3z3 - a4 + 5a4z2 - 5a4z4 + a4z6 - 2a5z + 6a5z3 - 5a5z5 + a5z7 - a6 + 6a6z2 - 5a6z4 + a6z6

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

{-4, 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=0 is the signature of 11116. 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 r = 4 j = 7                     1 j = 5                       j = 3                 1 1   j = 1             2 1       j = -1             1 1       j = -3         1 2 1         j = -5       1               j = -7       1 1             j = -9   1 1                 j = -11                       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, 116]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 116]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 14, 6, 15], X[7, 19, 8, 18], > X[9, 17, 10, 16], X[2, 11, 3, 12], X[20, 13, 21, 14], X[15, 22, 16, 1], > X[17, 9, 18, 8], X[12, 19, 13, 20], X[21, 7, 22, 6]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 116]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 11, -4, 9, -5, -2, 6, -10, 7, 3, -8, 5, -9, 4, 10, > -7, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 116]]
Out[5]=	DTCode[4, 10, -14, -18, -16, 2, 20, -22, -8, 12, -6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 116]][t]
Out[6]=	-2 2 3 - t - t
In[7]:=	Conway[Knot[11, NonAlternating, 116]][z]
Out[7]=	2 4 1 - 4 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 49], Knot[11, NonAlternating, 116]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 116]], KnotSignature[Knot[11, NonAlternating, 116]]}
Out[9]=	{1, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 116]][q]
Out[10]=	-6 -5 -4 -3 2 3 q - q + q - q + q - q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 116]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 116]][q]
Out[12]=	-20 -18 -12 -2 6 8 10 q + q - q - q + q + q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 116]][a, z]
Out[13]=	2 2 2 4 6 2 z 4 2 4 -2 + -- + a - a + a - 4 z + -- - a z - z 2 2 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 116]][a, z]
Out[14]=	2 2 2 4 6 5 2 6 z 2 2 4 2 -2 - -- - a - a - a + 2 a z - 2 a z + 8 z + ---- + a z + 5 a z + 2 2 a a 3 4 6 2 5 z 3 3 3 5 3 4 5 z 2 4 4 4 > 6 a z + ---- - 2 a z - a z + 6 a z - 6 z - ---- - a z - 5 a z - a 2 a 5 6 7 6 4 5 z 5 5 6 z 4 6 6 6 z 5 7 > 5 a z - ---- - 5 a z + z + -- + a z + a z + -- + a z a 2 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 116]], Vassiliev[3][Knot[11, NonAlternating, 116]]}
Out[15]=	{-4, 3}
In[16]:=	Kh[Knot[11, NonAlternating, 116]][q, t]
Out[16]=	-3 1 1 1 1 1 1 1 1 2 q + - + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 13 6 9 5 9 4 7 3 5 3 7 2 3 2 3 q t q t q t q t q t q t q t q t t 3 2 3 3 7 4 > - + q t + q t + q t + q t q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n116

K11n115 K11n117