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

K11a177 K11a179

Knotscape

This page is passe. Go here instead!    The Knot K11a178 Visit K11a178's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 18 2 22 20 10 8 6

Alexander Polynomial:

2t-3 - 11t-2 + 29t-1 - 39 + 29t - 11t2 + 2t3

Conway Polynomial:

1 + 3z2 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a294, ...}

Determinant and Signature:

{123, 2}

Jones Polynomial:

- q-2 + 4q-1 - 8 + 13q - 17q2 + 20q3 - 19q4 + 17q5 - 12q6 + 7q7 - 4q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- q-6 + 2q-4 - q-2 - 1 + 4q2 - 4q4 + 2q6 + q8 + 4q12 - 2q14 + 3q16 - q18 - 3q20 + 2q22 - 2q24 - q26 + q28

HOMFLY-PT Polynomial:

a-8z2 - 2a-6 - 3a-6z2 - 2a-6z4 + 3a-4 + 4a-4z2 + 2a-4z4 + a-4z6 + 2a-2z2 + 2a-2z4 + a-2z6 - z2 - z4

Kauffman Polynomial:

- 2a-10z4 + a-10z6 + 7a-9z3 - 11a-9z5 + 4a-9z7 - 4a-8z2 + 13a-8z4 - 16a-8z6 + 6a-8z8 - 5a-7z + 19a-7z3 - 17a-7z5 - a-7z7 + 4a-7z9 + 2a-6 - 16a-6z2 + 39a-6z4 - 39a-6z6 + 13a-6z8 + a-6z10 - 8a-5z + 23a-5z3 - 16a-5z5 - 6a-5z7 + 8a-5z9 + 3a-4 - 13a-4z2 + 29a-4z4 - 33a-4z6 + 14a-4z8 + a-4z10 - 4a-3z + 17a-3z3 - 22a-3z5 + 6a-3z7 + 4a-3z9 + a-2z2 - a-2z4 - 7a-2z6 + 7a-2z8 - a-1z + 5a-1z3 - 11a-1z5 + 7a-1z7 + 2z2 - 6z4 + 4z6 - az3 + az5

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

{3, 5}

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 11178. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                       1 j = 17                     3   j = 15                   4 1   j = 13                 8 3     j = 11               9 4       j = 9             10 8         j = 7           10 9           j = 5         7 10             j = 3       6 10               j = 1     3 8                 j = -1   1 5                   j = -3   3                     j = -5 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, 178]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 178]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[14, 5, 15, 6], X[16, 8, 17, 7], > X[18, 10, 19, 9], X[2, 12, 3, 11], X[22, 13, 1, 14], X[20, 16, 21, 15], > X[10, 18, 11, 17], X[8, 20, 9, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 178]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -3, 8, -4, 9, -5, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 178]]
Out[5]=	DTCode[4, 12, 14, 16, 18, 2, 22, 20, 10, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 178]][t]
Out[6]=	2 11 29 2 3 -39 + -- - -- + -- + 29 t - 11 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 178]][z]
Out[7]=	2 4 6 1 + 3 z + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 178], Knot[11, Alternating, 294]}
In[9]:=	{KnotDet[Knot[11, Alternating, 178]], KnotSignature[Knot[11, Alternating, 178]]}
Out[9]=	{123, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 178]][q]
Out[10]=	-2 4 2 3 4 5 6 7 8 9 -8 - q + - + 13 q - 17 q + 20 q - 19 q + 17 q - 12 q + 7 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 178]}
In[12]:=	A2Invariant[Knot[11, Alternating, 178]][q]
Out[12]=	-6 2 -2 2 4 6 8 12 14 16 18 -1 - q + -- - q + 4 q - 4 q + 2 q + q + 4 q - 2 q + 3 q - q - 4 q 20 22 24 26 28 > 3 q + 2 q - 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 178]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 6 -2 3 2 z 3 z 4 z 2 z 4 2 z 2 z 2 z z z -- + -- - z + -- - ---- + ---- + ---- - z - ---- + ---- + ---- + -- + -- 6 4 8 6 4 2 6 4 2 4 2 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 178]][a, z]
Out[14]=	2 2 2 2 3 2 3 5 z 8 z 4 z z 2 4 z 16 z 13 z z 7 z -- + -- - --- - --- - --- - - + 2 z - ---- - ----- - ----- + -- + ---- + 6 4 7 5 3 a 8 6 4 2 9 a a a a a a a a a a 3 3 3 3 4 4 4 4 19 z 23 z 17 z 5 z 3 4 2 z 13 z 39 z 29 z > ----- + ----- + ----- + ---- - a z - 6 z - ---- + ----- + ----- + ----- - 7 5 3 a 10 8 6 4 a a a a a a a 4 5 5 5 5 5 6 6 z 11 z 17 z 16 z 22 z 11 z 5 6 z 16 z > -- - ----- - ----- - ----- - ----- - ----- + a z + 4 z + --- - ----- - 2 9 7 5 3 a 10 8 a a a a a a a 6 6 6 7 7 7 7 7 8 8 39 z 33 z 7 z 4 z z 6 z 6 z 7 z 6 z 13 z > ----- - ----- - ---- + ---- - -- - ---- + ---- + ---- + ---- + ----- + 6 4 2 9 7 5 3 a 8 6 a a a a a a a a a 8 8 9 9 9 10 10 14 z 7 z 4 z 8 z 4 z z z > ----- + ---- + ---- + ---- + ---- + --- + --- 4 2 7 5 3 6 4 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 178]], Vassiliev[3][Knot[11, Alternating, 178]]}
Out[15]=	{3, 5}
In[16]:=	Kh[Knot[11, Alternating, 178]][q, t]
Out[16]=	3 1 3 1 5 3 q 3 5 5 2 8 q + 6 q + ----- + ----- + ---- + --- + --- + 10 q t + 7 q t + 10 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 10 q t + 9 q t + 10 q t + 8 q t + 9 q t + 4 q t + 8 q t + 13 6 15 6 15 7 17 7 19 8 > 3 q t + 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a178

K11a177 K11a179