K11a174

K11a173

K11a175

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a174 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,4,13,3 X14,5,15,6 X16,7,17,8 X18,10,19,9 X2,12,3,11 X22,13,1,14 X6,15,7,16 X20,17,21,18 X8,20,9,19 X10,21,11,22
Gauss code	1, -6, 2, -1, 3, -8, 4, -10, 5, -11, 6, -2, 7, -3, 8, -4, 9, -5, 10, -9, 11, -7
Dowker-Thistlethwaite code	4 12 14 16 18 2 22 6 20 8 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a174/ThurstonBennequinNumber Hyperbolic Volume 12.1343 A-Polynomial See Data:K11a174/A-polynomial

[edit Notes for K11a174's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 2

[edit Notes for K11a174's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-11t^{2}+15t-15+15t^{-1}-11t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 79, -2 }
Jones polynomial	$-q^{4}+3q^{3}-5q^{2}+8q-10+12q^{-1}-12q^{-2}+11q^{-3}-8q^{-4}+5q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-13a^{2}z^{4}-z^{4}a^{-2}+9z^{4}+4a^{4}z^{2}-12a^{2}z^{2}-3z^{2}a^{-2}+11z^{2}+a^{4}-3a^{2}-a^{-2}+4$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+6az^{9}+3z^{9}a^{-1}+4a^{4}z^{8}+5a^{2}z^{8}+3z^{8}a^{-2}+4z^{8}+4a^{5}z^{7}-4a^{3}z^{7}-19az^{7}-10z^{7}a^{-1}+z^{7}a^{-3}+4a^{6}z^{6}-5a^{4}z^{6}-24a^{2}z^{6}-13z^{6}a^{-2}-28z^{6}+3a^{7}z^{5}-2a^{5}z^{5}+a^{3}z^{5}+15az^{5}+5z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-4a^{6}z^{4}+3a^{4}z^{4}+31a^{2}z^{4}+16z^{4}a^{-2}+39z^{4}-4a^{7}z^{3}-3a^{5}z^{3}-a^{3}z^{3}-2az^{3}+4z^{3}a^{-1}+4z^{3}a^{-3}-a^{8}z^{2}+a^{6}z^{2}-2a^{4}z^{2}-17a^{2}z^{2}-7z^{2}a^{-2}-20z^{2}+a^{7}z+2a^{5}z+a^{3}z-az-2za^{-1}-za^{-3}+a^{4}+3a^{2}+a^{-2}+4$
The A2 invariant	$q^{20}-q^{18}+q^{16}-q^{14}-q^{12}+q^{10}-2q^{8}+3q^{6}-q^{4}+q^{2}+1-q^{-2}+2q^{-4}+q^{-8}-q^{-12}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+4q^{106}-2q^{104}-4q^{102}+12q^{100}-17q^{98}+22q^{96}-21q^{94}+10q^{92}+5q^{90}-22q^{88}+36q^{86}-43q^{84}+41q^{82}-27q^{80}+7q^{78}+20q^{76}-40q^{74}+56q^{72}-55q^{70}+45q^{68}-29q^{66}-q^{64}+29q^{62}-52q^{60}+63q^{58}-55q^{56}+33q^{54}-4q^{52}-30q^{50}+47q^{48}-45q^{46}+21q^{44}+11q^{42}-42q^{40}+48q^{38}-24q^{36}-19q^{34}+67q^{32}-95q^{30}+89q^{28}-45q^{26}-24q^{24}+92q^{22}-133q^{20}+133q^{18}-86q^{16}+15q^{14}+58q^{12}-105q^{10}+117q^{8}-88q^{6}+32q^{4}+25q^{2}-67+72q^{-2}-39q^{-4}-7q^{-6}+56q^{-8}-76q^{-10}+62q^{-12}-15q^{-14}-46q^{-16}+98q^{-18}-115q^{-20}+92q^{-22}-34q^{-24}-32q^{-26}+86q^{-28}-105q^{-30}+93q^{-32}-53q^{-34}+4q^{-36}+32q^{-38}-53q^{-40}+49q^{-42}-34q^{-44}+16q^{-46}+q^{-48}-10q^{-50}+10q^{-52}-9q^{-54}+5q^{-56}-2q^{-58}+q^{-60}$

Further Quantum Invariants

K11a174 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a174"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-t^{4}+5t^{3}-11t^{2}+15t-15+15t^{-1}-11t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-3z^{6}-z^{4}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 79, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{4}+3q^{3}-5q^{2}+8q-10+12q^{-1}-12q^{-2}+11q^{-3}-8q^{-4}+5q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-13a^{2}z^{4}-z^{4}a^{-2}+9z^{4}+4a^{4}z^{2}-12a^{2}z^{2}-3z^{2}a^{-2}+11z^{2}+a^{4}-3a^{2}-a^{-2}+4$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+6az^{9}+3z^{9}a^{-1}+4a^{4}z^{8}+5a^{2}z^{8}+3z^{8}a^{-2}+4z^{8}+4a^{5}z^{7}-4a^{3}z^{7}-19az^{7}-10z^{7}a^{-1}+z^{7}a^{-3}+4a^{6}z^{6}-5a^{4}z^{6}-24a^{2}z^{6}-13z^{6}a^{-2}-28z^{6}+3a^{7}z^{5}-2a^{5}z^{5}+a^{3}z^{5}+15az^{5}+5z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-4a^{6}z^{4}+3a^{4}z^{4}+31a^{2}z^{4}+16z^{4}a^{-2}+39z^{4}-4a^{7}z^{3}-3a^{5}z^{3}-a^{3}z^{3}-2az^{3}+4z^{3}a^{-1}+4z^{3}a^{-3}-a^{8}z^{2}+a^{6}z^{2}-2a^{4}z^{2}-17a^{2}z^{2}-7z^{2}a^{-2}-20z^{2}+a^{7}z+2a^{5}z+a^{3}z-az-2za^{-1}-za^{-3}+a^{4}+3a^{2}+a^{-2}+4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a174"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $-t^{4}+5t^{3}-11t^{2}+15t-15+15t^{-1}-11t^{-2}+5t^{-3}-t^{-4}$, $-q^{4}+3q^{3}-5q^{2}+8q-10+12q^{-1}-12q^{-2}+11q^{-3}-8q^{-4}+5q^{-5}-3q^{-6}+q^{-7}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(0, 1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $0$ $8$ $0$ $0$ $8$ $0$ $-{\frac {208}{3}}$ $-{\frac {160}{3}}$ $-24$ $0$ $32$ $0$ $0$ $208$ ${\frac {536}{3}}$ $-{\frac {104}{3}}$ ${\frac {112}{3}}$ $-64$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 K11a174. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 9                       1 -1 7                     2   2 5                   3 1   -2 3                 5 2     3 1               5 3       -2 -1             7 5         2 -3           6 6           0 -5         5 6             -1 -7       3 6               3 -9     2 5                 -3 -11   1 3                   2 -13   2                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a173 K11a175