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

K11n145 K11n147

Knotscape

This page is passe. Go here instead!    The Knot K11n146 Visit K11n146's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X14,4,15,3 X10,5,11,6 X20,8,21,7 X22,9,1,10 X11,19,12,18 X2,14,3,13 X8,15,9,16 X6,18,7,17 X19,13,20,12 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 5t-2 + 15t-1 - 21 + 15t - 5t2 + t3

Conway Polynomial:

1 + 4z2 + z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{K11n167, ...}

Determinant and Signature:

{63, 2}

Jones Polynomial:

- 1 + 4q - 7q2 + 10q3 - 10q4 + 11q5 - 9q6 + 6q7 - 4q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- 1 + 2q2 - 2q4 + 3q8 + 4q12 + q16 - 3q20 + q22 - 2q24 - q26 + q28

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-10z4 + a-10z6 + a-9z + 7a-9z3 - 12a-9z5 + 4a-9z7 - 4a-8z2 + 11a-8z4 - 15a-8z6 + 5a-8z8 - 4a-7z + 18a-7z3 - 20a-7z5 + 2a-7z7 + 2a-7z9 + 3a-6 - 18a-6z2 + 33a-6z4 - 29a-6z6 + 9a-6z8 - 8a-5z + 18a-5z3 - 11a-5z5 + 2a-5z9 + 5a-4 - 17a-4z2 + 24a-4z4 - 13a-4z6 + 4a-4z8 - 4a-3z + 8a-3z3 - 3a-3z5 + 2a-3z7 + a-2 - 3a-2z2 + 4a-2z4 - a-1z + a-1z3

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=2 is the signature of 11146. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj 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               3 1   j = 13             6 3     j = 11           5 3       j = 9         5 6         j = 7       5 5           j = 5     2 5             j = 3   2 5               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, NonAlternating, 146]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 146]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[10, 5, 11, 6], X[20, 8, 21, 7], > X[22, 9, 1, 10], X[11, 19, 12, 18], X[2, 14, 3, 13], X[8, 15, 9, 16], > X[6, 18, 7, 17], X[19, 13, 20, 12], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 146]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -9, 4, -8, 5, -3, -6, 10, 7, -2, 8, -11, 9, 6, -10, > -4, 11, -5]
In[5]:=	DTCode[Knot[11, NonAlternating, 146]]
Out[5]=	DTCode[4, 14, 10, 20, 22, -18, 2, 8, 6, -12, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 146]][t]
Out[6]=	-3 5 15 2 3 -21 + t - -- + -- + 15 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 146]][z]
Out[7]=	2 4 6 1 + 4 z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 146], Knot[11, NonAlternating, 167]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 146]], KnotSignature[Knot[11, NonAlternating, 146]]}
Out[9]=	{63, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 146]][q]
Out[10]=	2 3 4 5 6 7 8 9 -1 + 4 q - 7 q + 10 q - 10 q + 11 q - 9 q + 6 q - 4 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 146]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 146]][q]
Out[12]=	2 4 8 12 16 20 22 24 26 28 -1 + 2 q - 2 q + 3 q + 4 q + q - 3 q + q - 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 146]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -3 5 -2 z 4 z 8 z z 2 z 4 z z z -- + -- - a + -- - ---- + ---- - -- - ---- + ---- - -- + -- 6 4 8 6 4 2 6 4 2 4 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 146]][a, z]
Out[14]=	2 2 2 2 3 3 5 -2 z 4 z 8 z 4 z z 4 z 18 z 17 z 3 z 7 z -- + -- + a + -- - --- - --- - --- - - - ---- - ----- - ----- - ---- + ---- + 6 4 9 7 5 3 a 8 6 4 2 9 a a a a a a a a a a a 3 3 3 3 4 4 4 4 4 5 18 z 18 z 8 z z 2 z 11 z 33 z 24 z 4 z 12 z > ----- + ----- + ---- + -- - ---- + ----- + ----- + ----- + ---- - ----- - 7 5 3 a 10 8 6 4 2 9 a a a a a a a a a 5 5 5 6 6 6 6 7 7 7 20 z 11 z 3 z z 15 z 29 z 13 z 4 z 2 z 2 z > ----- - ----- - ---- + --- - ----- - ----- - ----- + ---- + ---- + ---- + 7 5 3 10 8 6 4 9 7 3 a a a a a a a a a a 8 8 8 9 9 5 z 9 z 4 z 2 z 2 z > ---- + ---- + ---- + ---- + ---- 8 6 4 7 5 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 146]], Vassiliev[3][Knot[11, NonAlternating, 146]]}
Out[15]=	{4, 7}
In[16]:=	Kh[Knot[11, NonAlternating, 146]][q, t]
Out[16]=	3 1 3 5 5 2 7 2 7 3 9 3 3 q + 2 q + --- + 5 q t + 2 q t + 5 q t + 5 q t + 5 q t + 5 q t + q t 9 4 11 4 11 5 13 5 13 6 15 6 15 7 > 6 q t + 5 q t + 3 q t + 6 q t + 3 q t + 3 q t + q t + 17 7 19 8 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n146

K11n145 K11n147