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

K11a224 K11a226

Knotscape

This page is passe. Go here instead!    The Knot K11a225 Visit K11a225's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 18 20 16 14 2 10 22 6 8

Alexander Polynomial:

- 2t-3 + 8t-2 - 11t-1 + 11 - 11t + 8t2 - 2t3

Conway Polynomial:

1 + 3z2 - 4z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{53, -4}

Jones Polynomial:

- q-9 + 2q-8 - 4q-7 + 6q-6 - 7q-5 + 8q-4 - 7q-3 + 7q-2 - 5q-1 + 3 - 2q + q2

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

{...}

A2 (sl(3)) Invariant:

- q-28 - q-26 - q-22 + q-20 + q-18 + q-16 + 2q-14 - q-12 + q-10 + q-4 - q-2 + q6

HOMFLY-PT Polynomial:

1 + 3z2 + z4 - a2 - 4a2z2 - 4a2z4 - a2z6 - a4z2 - 3a4z4 - a4z6 + 3a6 + 6a6z2 + 2a6z4 - 2a8 - a8z2

Kauffman Polynomial:

1 - 6z2 + 11z4 - 6z6 + z8 + 2az - 15az3 + 23az5 - 12az7 + 2az9 + a2 - 8a2z2 + 13a2z4 - 3a2z6 - 3a2z8 + a2z10 + 3a3z - 16a3z3 + 28a3z5 - 19a3z7 + 4a3z9 + a4z2 + 3a4z4 - 5a4z6 - a4z8 + a4z10 + 4a5z3 - 5a5z5 - 3a5z7 + 2a5z9 - 3a6 + 9a6z2 - 7a6z4 - 4a6z6 + 3a6z8 + a7z + 2a7z3 - 7a7z5 + 4a7z7 - 2a8 + 5a8z2 - 6a8z4 + 4a8z6 + a9z - 2a9z3 + 3a9z5 - a10z2 + 2a10z4 - a11z + a11z3

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

{3, -8}

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 11225. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 5                       1 j = 3                     1   j = 1                   2 1   j = -1                 3 1     j = -3               4 2       j = -5             4 4         j = -7           4 3           j = -9         3 4             j = -11       3 4               j = -13     1 3                 j = -15   1 3                   j = -17   1                     j = -19 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, 225]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 225]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[18, 5, 19, 6], X[20, 7, 21, 8], > X[16, 10, 17, 9], X[14, 12, 15, 11], X[2, 13, 3, 14], X[10, 16, 11, 15], > X[22, 17, 1, 18], X[6, 19, 7, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 225]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -10, 4, -11, 5, -8, 6, -2, 7, -6, 8, -5, 9, -3, 10, > -4, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 225]]
Out[5]=	DTCode[4, 12, 18, 20, 16, 14, 2, 10, 22, 6, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 225]][t]
Out[6]=	2 8 11 2 3 11 - -- + -- - -- - 11 t + 8 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 225]][z]
Out[7]=	2 4 6 1 + 3 z - 4 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 225]}
In[9]:=	{KnotDet[Knot[11, Alternating, 225]], KnotSignature[Knot[11, Alternating, 225]]}
Out[9]=	{53, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 225]][q]
Out[10]=	-9 2 4 6 7 8 7 7 5 2 3 - q + -- - -- + -- - -- + -- - -- + -- - - - 2 q + q 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 225]}
In[12]:=	A2Invariant[Knot[11, Alternating, 225]][q]
Out[12]=	-28 -26 -22 -20 -18 -16 2 -12 -10 -4 -2 6 -q - q - q + q + q + q + --- - q + q + q - q + q 14 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 225]][a, z]
Out[13]=	2 6 8 2 2 2 4 2 6 2 8 2 4 1 - a + 3 a - 2 a + 3 z - 4 a z - a z + 6 a z - a z + z - 2 4 4 4 6 4 2 6 4 6 > 4 a z - 3 a z + 2 a z - a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 225]][a, z]
Out[14]=	2 6 8 3 7 9 11 2 2 2 1 + a - 3 a - 2 a + 2 a z + 3 a z + a z + a z - a z - 6 z - 8 a z + 4 2 6 2 8 2 10 2 3 3 3 5 3 > a z + 9 a z + 5 a z - a z - 15 a z - 16 a z + 4 a z + 7 3 9 3 11 3 4 2 4 4 4 6 4 > 2 a z - 2 a z + a z + 11 z + 13 a z + 3 a z - 7 a z - 8 4 10 4 5 3 5 5 5 7 5 9 5 > 6 a z + 2 a z + 23 a z + 28 a z - 5 a z - 7 a z + 3 a z - 6 2 6 4 6 6 6 8 6 7 3 7 > 6 z - 3 a z - 5 a z - 4 a z + 4 a z - 12 a z - 19 a z - 5 7 7 7 8 2 8 4 8 6 8 9 3 9 > 3 a z + 4 a z + z - 3 a z - a z + 3 a z + 2 a z + 4 a z + 5 9 2 10 4 10 > 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 225]], Vassiliev[3][Knot[11, Alternating, 225]]}
Out[15]=	{3, -8}
In[16]:=	Kh[Knot[11, Alternating, 225]][q, t]
Out[16]=	4 4 1 1 1 3 1 3 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 11 4 q q q t q t q t q t q t q t q t 2 4 3 4 4 3 4 2 t 3 t t 2 > ------ + ----- + ----- + ----- + ---- + ---- + --- + --- + -- + 2 q t + 11 3 9 3 9 2 7 2 7 5 3 q q q t q t q t q t q t q t q 3 3 3 5 4 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a225

K11a224 K11a226