# 8 19

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 8 19's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 19 at Knotilus! 8 19 is the first non-obvious torus knot in the table - it is in fact T(4,3). It is also the pretzel knot P(3,3,-2).

8_19 is the first non-homologically thin knot in the Rolfsen table. (That is, it's the first knot whose Khovanov homology has 'off-diagonal' elements.)

 Knotscape Symmetrical form ; (3,4) torus knot True-lover's knot with sticked free ends Equal to the previous, from knotilus Pretzel knot P(2,-3,-3) French logo Seen in Singapore

### Knot presentations

 Planar diagram presentation X4251 X8493 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,1,12,16 X15,11,16,10 X2837 Gauss code 1, -8, 2, -1, -4, 5, 8, -2, -3, 7, -6, 4, -5, 3, -7, 6 Dowker-Thistlethwaite code 4 8 -12 2 -14 -16 -6 -10 Conway Notation [3,3,2-]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 8, width is 3,

Braid index is 3

[{4, 10}, {3, 5}, {1, 4}, {6, 9}, {5, 8}, {2, 6}, {10, 3}, {9, 7}, {8, 2}, {7, 1}]
 Knot 8_19. A graph which shows knot 8_19. A part of a knot and a part of a graph.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [5][-12] Hyperbolic Volume Not hyperbolic A-Polynomial See Data:8 19/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 3}$ Topological 4 genus ${\displaystyle 3}$ Concordance genus ${\displaystyle 3}$ Rasmussen s-Invariant -6

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{3}-t^{2}+1-t^{-2}+t^{-3}}$ Conway polynomial ${\displaystyle z^{6}+5z^{4}+5z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 3, 6 } Jones polynomial ${\displaystyle -q^{8}+q^{5}+q^{3}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{6}a^{-6}+6z^{4}a^{-6}-z^{4}a^{-8}+10z^{2}a^{-6}-5z^{2}a^{-8}+5a^{-6}-5a^{-8}+a^{-10}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{6}a^{-6}+z^{6}a^{-8}+z^{5}a^{-7}+z^{5}a^{-9}-6z^{4}a^{-6}-6z^{4}a^{-8}-5z^{3}a^{-7}-5z^{3}a^{-9}+10z^{2}a^{-6}+10z^{2}a^{-8}+5za^{-7}+5za^{-9}-5a^{-6}-5a^{-8}-a^{-10}}$ The A2 invariant ${\displaystyle q^{-10}+q^{-12}+2q^{-14}+2q^{-16}+2q^{-18}-q^{-22}-2q^{-24}-2q^{-26}-q^{-28}+q^{-32}}$ The G2 invariant ${\displaystyle q^{-50}+q^{-52}+q^{-54}+q^{-56}+q^{-58}+q^{-60}+2q^{-62}+2q^{-64}+q^{-66}+q^{-68}+2q^{-70}+2q^{-72}+2q^{-74}+q^{-76}+q^{-80}+2q^{-82}-q^{-94}-2q^{-96}-q^{-98}-q^{-100}-2q^{-102}-2q^{-104}-2q^{-106}-q^{-108}-q^{-110}-2q^{-112}-2q^{-114}-q^{-116}-q^{-122}-q^{-124}+q^{-126}+q^{-128}+q^{-136}+q^{-138}+q^{-144}}$

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (5, 10)
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 ${\displaystyle 20}$ ${\displaystyle 80}$ ${\displaystyle 200}$ ${\displaystyle {\frac {1270}{3}}}$ ${\displaystyle {\frac {170}{3}}}$ ${\displaystyle 1600}$ ${\displaystyle {\frac {7520}{3}}}$ ${\displaystyle {\frac {1280}{3}}}$ ${\displaystyle 272}$ ${\displaystyle {\frac {4000}{3}}}$ ${\displaystyle 3200}$ ${\displaystyle {\frac {25400}{3}}}$ ${\displaystyle {\frac {3400}{3}}}$ ${\displaystyle {\frac {91951}{6}}}$ ${\displaystyle {\frac {2818}{3}}}$ ${\displaystyle {\frac {44990}{9}}}$ ${\displaystyle {\frac {1429}{18}}}$ ${\displaystyle {\frac {3631}{6}}}$

V2,1 through V6,9 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 ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle s=}$6 is the signature of 8 19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
012345χ
17     1-1
15     1-1
13   11 0
11    1 1
9  1   1
71     1
51     1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=3}$ ${\displaystyle i=5}$ ${\displaystyle i=7}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=1}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$