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

K11a176 K11a178

Knotscape

This page is passe. Go here instead!    The Knot K11a177 Visit K11a177'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 X8,18,9,17 X10,20,11,19 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 18 2 22 20 8 10 6

Alexander Polynomial:

t-4 - 5t-3 + 13t-2 - 19t-1 + 21 - 19t + 13t2 - 5t3 + t4

Conway Polynomial:

1 + 4z2 + 3z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{97, 4}

Jones Polynomial:

- q-1 + 3 - 5q + 9q2 - 12q3 + 15q4 - 15q5 + 14q6 - 11q7 + 7q8 - 4q9 + q10

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

{...}

A2 (sl(3)) Invariant:

- q-2 + 1 - q2 + q4 + 2q6 - q8 + 4q10 - 2q12 + 2q14 + q16 - q18 + 2q20 - 3q22 - q26 - q28 + q30

HOMFLY-PT Polynomial:

2a-8z2 + a-8z4 - 3a-6 - 9a-6z2 - 8a-6z4 - 2a-6z6 + 5a-4 + 15a-4z2 + 14a-4z4 + 6a-4z6 + a-4z8 - a-2 - 4a-2z2 - 4a-2z4 - a-2z6

Kauffman Polynomial:

a-12z4 - 3a-11z3 + 4a-11z5 + a-10z2 - 7a-10z4 + 7a-10z6 + 2a-9z - a-9z3 - 8a-9z5 + 8a-9z7 + a-8z4 - 8a-8z6 + 7a-8z8 - a-7z + 8a-7z3 - 12a-7z5 + 4a-7z9 + 3a-6 - 18a-6z2 + 39a-6z4 - 36a-6z6 + 10a-6z8 + a-6z10 - 5a-5z + 8a-5z3 + 7a-5z5 - 18a-5z7 + 7a-5z9 + 5a-4 - 25a-4z2 + 47a-4z4 - 34a-4z6 + 6a-4z8 + a-4z10 - 3a-3z + 6a-3z3 + 3a-3z5 - 9a-3z7 + 3a-3z9 + a-2 - 8a-2z2 + 17a-2z4 - 13a-2z6 + 3a-2z8 - a-1z + 4a-1z3 - 4a-1z5 + a-1z7

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

{4, 7}

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=4 is the signature of 11177. 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 = 21                       1 j = 19                     3   j = 17                   4 1   j = 15                 7 3     j = 13               7 4       j = 11             8 7         j = 9           7 7           j = 7         5 8             j = 5       4 7               j = 3     2 6                 j = 1   1 3                   j = -1   2                     j = -3 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, 177]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 177]]
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[8, 18, 9, 17], X[10, 20, 11, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 177]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -9, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 177]]
Out[5]=	DTCode[4, 12, 14, 16, 18, 2, 22, 20, 8, 10, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 177]][t]
Out[6]=	-4 5 13 19 2 3 4 21 + t - -- + -- - -- - 19 t + 13 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 177]][z]
Out[7]=	2 4 6 8 1 + 4 z + 3 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 177]}
In[9]:=	{KnotDet[Knot[11, Alternating, 177]], KnotSignature[Knot[11, Alternating, 177]]}
Out[9]=	{97, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 177]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 10 3 - - - 5 q + 9 q - 12 q + 15 q - 15 q + 14 q - 11 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, 177]}
In[12]:=	A2Invariant[Knot[11, Alternating, 177]][q]
Out[12]=	-2 2 4 6 8 10 12 14 16 18 20 1 - q - q + q + 2 q - q + 4 q - 2 q + 2 q + q - q + 2 q - 22 26 28 30 > 3 q - q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 177]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 6 -3 5 -2 2 z 9 z 15 z 4 z z 8 z 14 z 4 z 2 z -- + -- - a + ---- - ---- + ----- - ---- + -- - ---- + ----- - ---- - ---- + 6 4 8 6 4 2 8 6 4 2 6 a a a a a a a a a a a 6 6 8 6 z z z > ---- - -- + -- 4 2 4 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 177]][a, z]
Out[14]=	2 2 2 2 3 3 5 -2 2 z z 5 z 3 z z z 18 z 25 z 8 z 3 z -- + -- + a + --- - -- - --- - --- - - + --- - ----- - ----- - ---- - ---- - 6 4 9 7 5 3 a 10 6 4 2 11 a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 4 z 8 z 8 z 6 z 4 z z 7 z z 39 z 47 z 17 z > -- + ---- + ---- + ---- + ---- + --- - ---- + -- + ----- + ----- + ----- + 9 7 5 3 a 12 10 8 6 4 2 a a a a a a a a a a 5 5 5 5 5 5 6 6 6 6 4 z 8 z 12 z 7 z 3 z 4 z 7 z 8 z 36 z 34 z > ---- - ---- - ----- + ---- + ---- - ---- + ---- - ---- - ----- - ----- - 11 9 7 5 3 a 10 8 6 4 a a a a a a a a a 6 7 7 7 7 8 8 8 8 9 13 z 8 z 18 z 9 z z 7 z 10 z 6 z 3 z 4 z > ----- + ---- - ----- - ---- + -- + ---- + ----- + ---- + ---- + ---- + 2 9 5 3 a 8 6 4 2 7 a a a a a a a a a 9 9 10 10 7 z 3 z z z > ---- + ---- + --- + --- 5 3 6 4 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 177]], Vassiliev[3][Knot[11, Alternating, 177]]}
Out[15]=	{4, 7}
In[16]:=	Kh[Knot[11, Alternating, 177]][q, t]
Out[16]=	3 3 5 1 2 q 3 q 2 q 5 7 7 2 6 q + 4 q + ----- + ---- + -- + --- + ---- + 7 q t + 5 q t + 8 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 15 5 > 7 q t + 7 q t + 8 q t + 7 q t + 7 q t + 4 q t + 7 q t + 15 6 17 6 17 7 19 7 21 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 K11a177

K11a176 K11a178