K11a171

K11a170

K11a172

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a171 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a171/ThurstonBennequinNumber Hyperbolic Volume 18.6712 A-Polynomial See Data:K11a171/A-polynomial

[edit Notes for K11a171'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 K11a171's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+7t^{3}-21t^{2}+39t-47+39t^{-1}-21t^{-2}+7t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-z^{6}+z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 183, 2 }
Jones polynomial	$-q^{8}+5q^{7}-12q^{6}+19q^{5}-26q^{4}+30q^{3}-29q^{2}+26q-18+11q^{-1}-5q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-4z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-5z^{4}a^{-2}+5z^{4}a^{-4}-z^{4}a^{-6}+2z^{4}+z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}+4a^{-2}-2a^{-4}-1$
Kauffman polynomial (db, data sources)	$3z^{10}a^{-2}+3z^{10}a^{-4}+9z^{9}a^{-1}+20z^{9}a^{-3}+11z^{9}a^{-5}+23z^{8}a^{-2}+29z^{8}a^{-4}+16z^{8}a^{-6}+10z^{8}+5az^{7}-7z^{7}a^{-1}-23z^{7}a^{-3}+z^{7}a^{-5}+12z^{7}a^{-7}+a^{2}z^{6}-65z^{6}a^{-2}-71z^{6}a^{-4}-23z^{6}a^{-6}+5z^{6}a^{-8}-21z^{6}-9az^{5}-15z^{5}a^{-1}-18z^{5}a^{-3}-28z^{5}a^{-5}-15z^{5}a^{-7}+z^{5}a^{-9}-a^{2}z^{4}+47z^{4}a^{-2}+47z^{4}a^{-4}+12z^{4}a^{-6}-3z^{4}a^{-8}+14z^{4}+4az^{3}+14z^{3}a^{-1}+21z^{3}a^{-3}+17z^{3}a^{-5}+6z^{3}a^{-7}-6z^{2}a^{-2}-8z^{2}a^{-4}-4z^{2}a^{-6}-2z^{2}-za^{-1}-za^{-3}-za^{-5}-za^{-7}-4a^{-2}-2a^{-4}-1$
The A2 invariant	$q^{8}-3q^{6}+3q^{4}-3q^{2}-1+6q^{-2}-4q^{-4}+8q^{-6}-2q^{-8}+q^{-10}+q^{-12}-6q^{-14}+4q^{-16}-3q^{-18}+2q^{-22}-q^{-24}$
The G2 invariant	$q^{46}-4q^{44}+11q^{42}-24q^{40}+36q^{38}-43q^{36}+30q^{34}+20q^{32}-100q^{30}+210q^{28}-303q^{26}+316q^{24}-204q^{22}-78q^{20}+476q^{18}-859q^{16}+1062q^{14}-921q^{12}+376q^{10}+457q^{8}-1316q^{6}+1850q^{4}-1784q^{2}+1060+97q^{-2}-1270q^{-4}+1969q^{-6}-1893q^{-8}+1078q^{-10}+148q^{-12}-1230q^{-14}+1686q^{-16}-1303q^{-18}+249q^{-20}+1016q^{-22}-1900q^{-24}+2001q^{-26}-1197q^{-28}-222q^{-30}+1715q^{-32}-2695q^{-34}+2772q^{-36}-1870q^{-38}+304q^{-40}+1366q^{-42}-2538q^{-44}+2797q^{-46}-2076q^{-48}+684q^{-50}+801q^{-52}-1800q^{-54}+1931q^{-56}-1212q^{-58}+9q^{-60}+1116q^{-62}-1648q^{-64}+1352q^{-66}-394q^{-68}-811q^{-70}+1726q^{-72}-1967q^{-74}+1468q^{-76}-447q^{-78}-674q^{-80}+1480q^{-82}-1735q^{-84}+1427q^{-86}-747q^{-88}-13q^{-90}+601q^{-92}-883q^{-94}+847q^{-96}-591q^{-98}+270q^{-100}+17q^{-102}-196q^{-104}+252q^{-106}-230q^{-108}+152q^{-110}-71q^{-112}+14q^{-114}+22q^{-116}-31q^{-118}+28q^{-120}-20q^{-122}+10q^{-124}-4q^{-126}+q^{-128}$

Further Quantum Invariants

K11a171 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["K11a171"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+7t^{3}-21t^{2}+39t-47+39t^{-1}-21t^{-2}+7t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-z^{6}+z^{4}+2z^{2}+1$

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]=

{ 183, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+5q^{7}-12q^{6}+19q^{5}-26q^{4}+30q^{3}-29q^{2}+26q-18+11q^{-1}-5q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-4z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-5z^{4}a^{-2}+5z^{4}a^{-4}-z^{4}a^{-6}+2z^{4}+z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}+4a^{-2}-2a^{-4}-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3z^{10}a^{-2}+3z^{10}a^{-4}+9z^{9}a^{-1}+20z^{9}a^{-3}+11z^{9}a^{-5}+23z^{8}a^{-2}+29z^{8}a^{-4}+16z^{8}a^{-6}+10z^{8}+5az^{7}-7z^{7}a^{-1}-23z^{7}a^{-3}+z^{7}a^{-5}+12z^{7}a^{-7}+a^{2}z^{6}-65z^{6}a^{-2}-71z^{6}a^{-4}-23z^{6}a^{-6}+5z^{6}a^{-8}-21z^{6}-9az^{5}-15z^{5}a^{-1}-18z^{5}a^{-3}-28z^{5}a^{-5}-15z^{5}a^{-7}+z^{5}a^{-9}-a^{2}z^{4}+47z^{4}a^{-2}+47z^{4}a^{-4}+12z^{4}a^{-6}-3z^{4}a^{-8}+14z^{4}+4az^{3}+14z^{3}a^{-1}+21z^{3}a^{-3}+17z^{3}a^{-5}+6z^{3}a^{-7}-6z^{2}a^{-2}-8z^{2}a^{-4}-4z^{2}a^{-6}-2z^{2}-za^{-1}-za^{-3}-za^{-5}-za^{-7}-4a^{-2}-2a^{-4}-1$

"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["K11a171"];

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]=

{ $-t^{4}+7t^{3}-21t^{2}+39t-47+39t^{-1}-21t^{-2}+7t^{-3}-t^{-4}$, $-q^{8}+5q^{7}-12q^{6}+19q^{5}-26q^{4}+30q^{3}-29q^{2}+26q-18+11q^{-1}-5q^{-2}+q^{-3}$ }

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:

(2, 2)

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 $8$ $16$ $32$ ${\frac {172}{3}}$ ${\frac {20}{3}}$ $128$ ${\frac {544}{3}}$ ${\frac {64}{3}}$ $16$ ${\frac {256}{3}}$ $128$ ${\frac {1376}{3}}$ ${\frac {160}{3}}$ ${\frac {10471}{15}}$ ${\frac {1636}{15}}$ ${\frac {5404}{45}}$ ${\frac {89}{9}}$ $-{\frac {89}{15}}$

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 K11a171. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     4   4 13                   8 1   -7 11                 11 4     7 9               15 8       -7 7             15 11         4 5           14 15           1 3         12 15             -3 1       7 15               8 -1     4 11                 -7 -3   1 7                   6 -5   4                     -4 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=2$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{15}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{15}$ $r=4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a170 K11a172