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

K11a275 K11a277

Knotscape

This page is passe. Go here instead!    The Knot K11a276 Visit K11a276's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6271 X10,4,11,3 X14,6,15,5 X20,8,21,7 X4,10,5,9 X18,12,19,11 X2,14,3,13 X22,15,1,16 X8,18,9,17 X12,20,13,19 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 14 20 4 18 2 22 8 12 16

Alexander Polynomial:

- 3t-3 + 17t-2 - 37t-1 + 47 - 37t + 17t2 - 3t3

Conway Polynomial:

1 + 4z2 - z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{161, 4}

Jones Polynomial:

1 - 4q + 10q2 - 16q3 + 23q4 - 26q5 + 26q6 - 23q7 + 17q8 - 10q9 + 4q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 - 2q2 + q4 + 3q6 - 4q8 + 6q10 - q12 + 3q16 - 5q18 + 4q20 - 4q22 + q24 + 3q26 - 4q28 + 2q30 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 2a-8 + 5a-8z2 + 3a-8z4 - 3a-6 - 5a-6z2 - 5a-6z4 - 2a-6z6 + 3a-4 + 4a-4z2 - a-4z6 + a-2z2 + a-2z4

Kauffman Polynomial:

- a-13z3 + a-13z5 + a-12z2 - 4a-12z4 + 4a-12z6 - 3a-11z + 9a-11z3 - 13a-11z5 + 9a-11z7 + a-10 - 5a-10z2 + 12a-10z4 - 18a-10z6 + 12a-10z8 - 3a-9z + 15a-9z3 - 16a-9z5 - 3a-9z7 + 9a-9z9 + 2a-8 - 14a-8z2 + 39a-8z4 - 48a-8z6 + 18a-8z8 + 3a-8z10 - a-7z + 2a-7z3 + 8a-7z5 - 28a-7z7 + 17a-7z9 + 3a-6 - 17a-6z2 + 40a-6z4 - 45a-6z6 + 14a-6z8 + 3a-6z10 - a-5z + a-5z3 + 2a-5z5 - 12a-5z7 + 8a-5z9 + 3a-4 - 8a-4z2 + 15a-4z4 - 18a-4z6 + 8a-4z8 + 4a-3z3 - 8a-3z5 + 4a-3z7 + a-2z2 - 2a-2z4 + a-2z6

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

{4, 9}

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 11276. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                       1 j = 21                     3   j = 19                   7 1   j = 17                 10 3     j = 15               13 7       j = 13             13 10         j = 11           13 13           j = 9         10 13             j = 7       6 13               j = 5     4 10                 j = 3   1 7                   j = 1   3                     j = -1 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, 276]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 276]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 8, 21, 7], > X[4, 10, 5, 9], X[18, 12, 19, 11], X[2, 14, 3, 13], X[22, 15, 1, 16], > X[8, 18, 9, 17], X[12, 20, 13, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 276]]
Out[4]=	GaussCode[1, -7, 2, -5, 3, -1, 4, -9, 5, -2, 6, -10, 7, -3, 8, -11, 9, -6, 10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 276]]
Out[5]=	DTCode[6, 10, 14, 20, 4, 18, 2, 22, 8, 12, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 276]][t]
Out[6]=	3 17 37 2 3 47 - -- + -- - -- - 37 t + 17 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 276]][z]
Out[7]=	2 4 6 1 + 4 z - z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 276]}
In[9]:=	{KnotDet[Knot[11, Alternating, 276]], KnotSignature[Knot[11, Alternating, 276]]}
Out[9]=	{161, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 276]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 4 q + 10 q - 16 q + 23 q - 26 q + 26 q - 23 q + 17 q - 10 q + 10 11 > 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 276]}
In[12]:=	A2Invariant[Knot[11, Alternating, 276]][q]
Out[12]=	2 4 6 8 10 12 16 18 20 22 1 - 2 q + q + 3 q - 4 q + 6 q - q + 3 q - 5 q + 4 q - 4 q + 24 26 28 30 34 > q + 3 q - 4 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 276]][a, z]
Out[13]=	2 2 2 2 2 4 4 4 -10 2 3 3 z 5 z 5 z 4 z z 3 z 5 z z -a + -- - -- + -- - --- + ---- - ---- + ---- + -- + ---- - ---- + -- - 8 6 4 10 8 6 4 2 8 6 2 a a a a a a a a a a a 6 6 2 z z > ---- - -- 6 4 a a
In[14]:=	Kauffman[Knot[11, Alternating, 276]][a, z]
Out[14]=	2 2 2 2 2 -10 2 3 3 3 z 3 z z z z 5 z 14 z 17 z 8 z a + -- + -- + -- - --- - --- - -- - -- + --- - ---- - ----- - ----- - ---- + 8 6 4 11 9 7 5 12 10 8 6 4 a a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 4 z z 9 z 15 z 2 z z 4 z 4 z 12 z 39 z 40 z > -- - --- + ---- + ----- + ---- + -- + ---- - ---- + ----- + ----- + ----- + 2 13 11 9 7 5 3 12 10 8 6 a a a a a a a a a a a 4 4 5 5 5 5 5 5 6 6 15 z 2 z z 13 z 16 z 8 z 2 z 8 z 4 z 18 z > ----- - ---- + --- - ----- - ----- + ---- + ---- - ---- + ---- - ----- - 4 2 13 11 9 7 5 3 12 10 a a a a a a a a a a 6 6 6 6 7 7 7 7 7 8 48 z 45 z 18 z z 9 z 3 z 28 z 12 z 4 z 12 z > ----- - ----- - ----- + -- + ---- - ---- - ----- - ----- + ---- + ----- + 8 6 4 2 11 9 7 5 3 10 a a a a a a a a a a 8 8 8 9 9 9 10 10 18 z 14 z 8 z 9 z 17 z 8 z 3 z 3 z > ----- + ----- + ---- + ---- + ----- + ---- + ----- + ----- 8 6 4 9 7 5 8 6 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 276]], Vassiliev[3][Knot[11, Alternating, 276]]}
Out[15]=	{4, 9}
In[16]:=	Kh[Knot[11, Alternating, 276]][q, t]
Out[16]=	3 3 5 1 3 q q 5 7 7 2 9 2 7 q + 4 q + ---- + --- + -- + 10 q t + 6 q t + 13 q t + 10 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 > 13 q t + 13 q t + 13 q t + 13 q t + 10 q t + 13 q t + 15 6 17 6 17 7 19 7 19 8 21 8 23 9 > 7 q t + 10 q t + 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a276

K11a275 K11a277